I remember, as a student, being very struck by a poster arguing against an attempt by Edward Heath's government to bring in museum charges. It said: "We the undersigned oppose the introduction of admission charges" and carried the signatures of Van Gogh, Titian, Turner and some 50 other great artists. It made me realise a simple truth: that free admission is all about giving everyone, no matter what their means, the chance to see the greatest works of art, science and history that our nation has.
Over the following three decades, charges were indeed brought in. Some national collections valiantly held out against the tide; but most succumbed to charging, and in some cases the charges were high. To bring a family to the Science Museum or Natural History Museum became a substantial financial undertaking.
When I became Secretary of State in 1997, I was determined to change this. I believed passionately that these great treasure houses belonged to us all, and should be available for free, for ever. It took me four years to achieve that: convincing reluctant colleagues; securing additional funding; persuading some museum directors; achieving the removal of VAT. It was worth it, though; and the surge in visitor numbers – up by 150% over the last decade – has proved it.
On the day free admission began, 10 years ago, I was invited to cut the ribbon and throw open the doors at the Science Museum. About half an hour later, I was standing in the foyer, and a man approached me, carrying his daughter on his shoulders. He looked up at her and said, "I want you to say thank you to this man. It's because of him we're able to be here today." That, too, made it worth it. Chris Smith
Nicholas Serota, director, Tate
One of the great atrocities of the Spanish civil war was the bombing of the Basque town of Guernica by the German air force, lending their support to the Nationalist forces of General Franco. Picasso responded to the massacre by painting the vast mural Guernica, and for months afterwards made subsidiary paintings based on one of the figures in the mural: a weeping woman holding her dead child. Weeping Woman in Tate's collection is the last and most elaborate of the series. A portrait of Picasso's mistress, Dora Maar, the painting is an extraordinary depiction of female grief and a metaphor for a Spanish tragedy.
The Sir John Soane's Museum is one of my favourite small museums. The eight paintings in William Hogarth's A Rake's Progress series can rightly be described as one of the great masterpieces of British art. Created by Hogarth, the great 18th-century painter, engraver and satirist, they give us an acute glimpse into London life of the period, and the antics of its faded aristocracy and nouveau riche. The paintings were originally hung at Soane's country villa, but were moved back to Lincoln's Inn Fields (now the Soane museum) in 1810. They were hung in a new picture room at the rear of the house, where they remain today.
Joan Bakewell, broadcaster
If they start charging for museums I will go spare with rage; it's been a great leap forward. The Cholmondeley Ladies (1600-10) at Tate Britain is one of my favourites to drop in and see. I take the grandchildren to visit them all the time. Everyone can relate to it: they're like reassuring family friends – "Let's go and visit the Cholmondeley sisters." It's so lovely how very different from each other they are, but how much the same. You get pleasure from them: they're women; they're siblings; they look beautiful; they're a reflection of an earlier time; they're all the very simple things you enjoy in a painting. And they must have their own stories: the painting is full of possibilities.
Nicholas Penny, director, National Gallery
The painting most appropriate for this particular anniversary is Rubens's Peace and War, the proper title of which is Minerva Protects Pax from Mars – though I hate that title. It was a diplomatic gift from Rubens to Charles I, when the painter was acting as an envoy for Philip IV, but nevertheless seems to me a painting for everyone. It is allegory, it's portraiture, it's animal painting, it's fruit-and-vegetable painting, it's got quite a lot of landscape, it's got the female nude, it's got men in armour. It was a gift from the Duke of Sutherland to the newly founded National Gallery about 200 years after it was painted, an amazing gesture of support: the Duke was donating one of the most valuable paintings in London.
The work of art I always visit when I go to the Victoria and Albert museum is a white jade cup that is known to have been used by Shah Jahan, one of the great Moghul rulers of India. Curiously, it's not that different in date from the Rubens: the middle of the 17th century. It reminds me of the game animal, vegetable or mineral. It shows the transparency as well as the hardness of jade, but at the same time incorporates animal and vegetable: the lotus flower at the foot, and the head of an ibex, which forms the handle. It epitomises the art of so many different cultures although it's a quintessential, high quality product of Islamic civilisation. This and the Rubens are two pieces of court culture completely accessible to the man in the street.
Lauren Laverne, broadcaster
At the V&A Museum of Childhood in east London there is a Dutch doll's house from the 1600s. It's very beautiful and the craftsmanship that went into it is mindblowing. It's also interesting because it shows how a house ran at that time, in that place. The idea behind it was to teach little girls how to become wives; it illustrates how much of our culture is indoctrinated into us through play and leisure.
Whenever I go into the British Museum, that ceiling in the atrium makes you look up, and as soon as you look up like that you're like a kid again. It puts you into an inquisitive, exploratory frame of mind. That's what I like about the Museum of Childhood, too: it's a lovely blend of history, mystery and fun.
Neil MacGregor, director, British Museum
The Great God of War, a colossal statue of a lion, from Nimrud (northern Iraq), 883 – 859BC, would be my pick from the British museum. This figure acted as a guardian to the entrance of the temple of Ishtar. It is an example of the extraordinary achievement of creating in stone the noise of war as well as its savagery.
At the other end of the scale of human experience to the Assyrian tiger, Diego Velasquez's An Old Woman Cooking Eggs (1618) from the National Gallery of Scotland embodies the still mystery of domestic life. It's one of Velázquez's "bodegónes", a kitchen scene, and was painted when he was still a very young man, only 18 or 19.
John Leighton, director, National Galleries of Scotland
The portrait of Robert Burns by Alexander Nasmyth, made in 1787, is probably the best-known portrait in our collection, which opens free to the public today. Like many people, I saw it first on a shortbread tin, but when you come face to face with the original it's astonishingly vivid, and you can feel that spirit of democracy and generosity Burns is famed for. The artist left it unfinished because he was afraid to lose what he said was a superb likeness. In among all the grand, eloquent portraits of powerful people in this gallery, this small, modest picture speaks very loudly indeed.
My favourite work in another gallery is from the National in London. It's another portrait, this time by Jacques-Louis David of Jacobus Blauw, a young Dutch ambassador trying to negotiate a peace treaty with France. It's a very direct rendering of the clarity and youthful idealism people associated with the French revolution. If you imagine that this is 1795, with guillotines crashing all over the place, you'd have to be a particularly skilled diplomat to negotiate with the revolutionary government. The portrait gives no indication of that hardship – instead, you're drawn in by the rendering of the materials, the steely blue jacket with a hint of his hair powder on the collar, and this pink face that engages you so directly you feel you've come eye-to-eye with this young Dutchman.
AL Kennedy, author
Kelvingrove Art Gallery and Museum in Glasgow has a wonderful full-length statue of Robert Louis Stevenson. He's a writer I hugely admire, and the piece seems to catch something of his spirit in a way photographs of him don't. When I first moved to Glasgow and was very much a tyro writer, I would occasionally wander off to Kelvingrove and potter. The building was – and is – beautifully uplifting in itself, and much warmer than I could afford to keep my flat. I would always end up spending a while with the RLS statue. It's not idealised like his memorial in Saint Giles, or the standard depictions of the great and good; he looks like someone who thought and travelled and had a lean kind of energy and efficiency about him. I find it inspirational.
Christopher Brown, director, Ashmolean Museum
I'm going to be a little opportunistic and choose an object from our new Egyptian galleries. I'm hugely moved by a remarkable mace head we have that dates from 3,000 BC and comes from Hierakonpolis. It's called the scorpion mace head and depicts an emperor. He's wearing the crown of Upper Egypt and he's hunting. I'm just enormously impressed by its sophistication as a piece of early sculpture: eat your heart out, Donatello.
I worked for many years at the National in London. I particularly love a late painting there by Giovanni Bellini, the Madonna of the Meadow. It shows the virgin with Christ in her lap, but it's a premonition of the Pietà. It has a beautiful, desolate landscape on the left, and on the right a prosperous landscape with a beautiful view of the area north of Venice where Bellini was working. When I worked at the National, one of the great joys was that people would drop in to the gallery between trains at Charing Cross, to come in and see something. You don't feel: "Well, I've spent £5 – I've got to make it worth my while." You can just go and look at a single picture. That to me is the key to free admission.
Michael Dixon, director, Natural History Museum
The Archaeopteryx lithographica is the most valuable single fossil in our collection. It is a famous snapshot of evolution in action that demonstrates conclusively that modern birds are descended from dinosaurs. It has huge scientific, historical and financial value. Elsewhere, the Rosetta Stone at the British Museum is fundamentally important to our understanding of archaeology and ancient cultures.
Julia Peyton-Jones, co-director, Serpentine Gallery
We don't have a permanent collection, but the Serpentine Pavilion series, now in its 11th year, allows the public to enjoy the work of international architects who haven't yet completed a building in the UK, for three months over the summer. My favourite pavilion? I couldn't possibly say!
The Turbine Hall commissions at Tate Modern are an extraordinary rollcall of some of the greatest practitioners of today. Louise Bourgeois's spider, Maman, and I Do, I Undo, I Redo, launched the whole programme in the most remarkable way. It's wonderful to see how artists address that space.
Sandy Nairne, director, National Portrait Gallery
One of our most enigmatic portraits is the famous Chandos portrait of Shakespeare by John Taylor of 1610. It's wonderfully mysterious. Taylor is not an artist we know a great deal about, and there's been plenty of speculation as to whether this was taken from life. For me, the idea of why we look at portraits of figures in history is embodied in this picture. I look at it at least two or three times a week. We refer to it as our No 1 because it is the first portrait that entered the collection in 1856, given by Lord Earlsmere.
I often go and look at Rembrandt's Self-Portrait at the Age of 34 next door in the National Gallery. It's not very different in date from the Shakespeare – it's 1640 – but is the absolute complement: whereas the Taylor is all about Shakespeare, this is about Rembrandt himself. Like all great self-portraits, it makes you question who you are and absolutely crosses time – that sense of self-examination. It's just the most brilliant painting, and to be able to just walk in and look at it is a fabulous thing.
Iwona Blazwick, director, Whitechapel Gallery
One of my favourite exhibits currently on display here is the Bloomberg Commission: Josiah McElheny: The Past Was a Mirage I'd Left Far Behind. Large-scale, mirrored sculptures are arranged as multiple reflective screens for the artist's interpretation of experimental abstract films. What is interesting is the way McElheny has responded to the site, which was previously the reading room of the former Whitechapel Library, a haven for early modernist thinkers such as Isaac Rosenberg. The library was built as a "lantern for learning"; McElheny has used the moving images and illumination as central motifs.
Elsewhere, the Sir John Soane's Museum is a delight, with important works from Hogarth to Canaletto set among drawings, historical architectural models and other fascinating antiquities.
Martin Roth, director, Victoria and Albert museum
Our medieval and Renaissance galleries opened two years ago to house one of the world's most remarkable collections of treasures from the period, marking the end of the first phase of our plan to modernise the museum. They host the greatest collection of Italian Renaissance sculpture outside Italy, with an exceptional group of sculptures by Donatello who was the greatest sculptor of his time. I particularly admire this pieceThe Ascension with Christ Giving the Keys to St Peter (1428-30) which combines two different scenes from the Gospels – one of the finest surviving examples of his astonishing low-relief carving technique.
Gauguin was one of the most important artists of the 19th century, and his experimentations with new styles and radical expression continue to inspire people today. Vision after the Sermon is one of the masterpieces from the Scottish National Gallery collection and this dramatic work changed the course of the history of art. Gauguin travelled the world and it's fascinating to see the influence of many forms of art in his work, from Japanese prints to ceramics.