This week's best things to do in London, from the ATP finals to the Lord Mayor's Show ATP World Tour Finals, Greenwich The ATP is the Association of Tennis Professionals and its annual world tour concludes again this year at London's O2 Arena. As ever, the ATP World Tour Finals sees the world's top eight players in men's tennis round off the season by battling each other to be crowned the definitive tennis champion of 2012. For their trouble there's the chance to pick up a helping of the multimillion-dollar prize fund. With egos to bolster and serious money to be made, the competition is reliably fierce so expect star turns from Novak Djokovic, Andy Murray, Roger Federer, David Ferrer, Tomas Berdych, Jo-Wilfried Tsonga, Janko Tipsarevic and Mardy Fish. When: Nov 5-12 Where: The O2, Peninsula Square, London, SE10 0DX Tube: North Greenwich How much: tickets cost from £20 The Lord Mayor's Show, City of London The Lord Mayor's Show has been in existence for almost eight centuries. It originated in 1213, when King John awarded the City of London the right to choose its own mayor if the mayor travelled upriver to Westminster to pledge his loyalty to the Crown. On Saturday the show will trace a three-mile route to the Royal Courts, and the mayor will pledge his allegiance with over 6000 participants for company - expect to see marching bands, acrobats and military personnel joining him along the way, as well as 125 horses, 18 21 carriages, a Japanese Taiko drum band and a Sherman tank. Previous years have seen the show culminate in a huge fireworks display but following the massive Jubilee and Olympic displays that element has been curtailed this year. When: Nov 10 Where: Throughout the City of London. The website has more exact details and travel advice. How much: Free to watch London Jazz Festival If you've already experienced the annual London Jazz Festival you'll know it regularly brings established jazz stars to London, and provides a platform to emerging stars. This year the ten-day festival hopes to consolidate its growing reputation by hosting performances from artists such as Kurt Elling at the Southbank Centre, Dee Dee Bridgewater at Ronnie Scott's Jazz Club and Sonny Rollins at Barbican. When: November 9-18 London Ice Rinks Another sure sign of winter's approach is the opening of London's seasonal ice rinks each November. The process has now begun and this week you can take to the ice at the Natural History Museum and Canary Wharf. Major ice-rink openings to take place scheduled to take place later this month include those at Somerset House, the London Eye and the Tower of London.