What books must every math undergraduate read? I'm still a student, but the same books keep getting named by my tutors (Rudin, Royden).

I've read Baby Rudin and begun Royden though I'm unsure if there are other books that I "should" be working on if I want to study beyond Masters. I'm not there yet as I'm on a four year course and had a gap year between Years 3 and 4.

Please recommend for Algebra, Linear Algebra and Categories - Analysis, Set Theory, Measure theory (an area I have seen too little books dedicated for).

E.g. Spivak is very good for self learning basic real analysis, but Rudin really cuts to the heart.
EDIT: I now think that this list is long enough that I shall be maintaining it over time--updating it whenever I use a new book/learn a new subject. While every suggestion below should be taken with a grain of salt--I will say that I spend a huge amount of time sifting through books to find the ones that conform best to my (and hopefully your!) learning style.



Here is my two cents (for whatever that's worth). I tried to include all the topics I could imagine you could want to know at this point. I hope I picked the right level of difficult. Feel absolutely free to ask my specific opinion about any book.

Basic Analysis: Rudin--Apostol

Measure Theory: Royden (only if you get the newest fourth edition)--Folland

General Algebra: D&amp;F--Rotman--Lang--Grillet

Finite Group Theory: Isaacs-- Kurzweil

General Group Theory: Robinson--Rotman

Ring Theory: T.Y. Lam-- times two

Commutative Algebra: Eisenbud--A&amp;M--Reid

Homological Algebra: Weibel--Rotman--Vermani

Category Theory: Mac Lane--Adamek et. al--Berrick et. al--Awodey--Mitchell

Linear Algebra: Roman--Hoffman and Kunze--Golan

Field Theory: Morandi--Roman

Complex Analysis: Ahlfors--Cartan--Freitag

Riemann Surfaces: Varolin(great first read, can be a little sloppy though)--Freitag(overall great book for a second course in complex analysis!)--Forster(a little more old school, and with a slightly more algebraic bend then a differential geometric one)--Donaldson

SCV: Gunning et. al--Ebeling

Point-set Topology: Munkres--Steen et. al--Kelley

Differential Topology: Pollack et. al--Milnor--Lee

Algebraic Topology: Bredon--May-- Bott and Tu (great, great book)--Rotman--Massey--Tom Dieck

Differential Geometry: Do Carmo--Spivak--Jost--Lee

Representation Theory of Finite Groups: Serre--Steinberg--Liebeck--Isaacs

General Representation Theory: Fulton and Harris--Humphreys--Hall

Representation Theory of Compact Groups: Tom Dieck et. al--Sepanski

(Linear) Algebraic Groups: Springer--Humphreys

"Elementary" Number Theory: Niven et. al--Ireland et. al

Algebraic Number Theory: Ash--Lorenzini--Neukirch--Marcus--Washington

Fourier Analysis--Katznelson

Modular Forms: Diamond and Shurman--Stein

Local Fields:


Lorenz and Levy--Read chapters 23,24,25. This is by far my favorite quick reference, as well as "learning text" for the basics of local fields one needs to break into other topics (e.g. class field theory).
Serre--This is the classic book. It is definitely low on the readability side, especially notationally. It also has a tendency to consider things in more generality than is needed at a first go. This isn't bad, but is not good if you're trying to "brush up" or quickly learn local fields for another subject.
Fesenko et. al--A balance between 1. and 2. Definitely more readable than 2., but more comprehensive than 1. If you are wondering whether or not so-and-so needs Henselian, this is the place I'd check.
Iwasawa--A great place to learn the bare-bones of what one might need to learn class field theory. I am referencing, in particular, the first three chapters. If you are dead-set on JUST learning what you need to, this is a pretty good reference, but if you're likely to wonder about why so-and-so theorem is true, or get a broader understanding of the basics of local fields, I recommend 1. 


Class Field Theory: 


Lorenz and Levy--Read chapters 28-32, second only to Iwasawa, but with a different flavor (cohomological vs. formal group laws)
Tate and Artin--The classic book. A little less readable then any of the alternatives here.
Childress--Focused mostly on the global theory opposed to the local. Actually deduces local at the end as a result of global. Thus, very old school.
Iwasawa (read the rest of it!)
Milne--Where I first started learning it. Very good, but definitely roughly hewn. A lot of details are left out, and he sometimes forgets to tell you where you are going.


Metric Groups: Markley

Algebraic Geometry: Reid--Shafarevich--Hartshorne--Griffiths and Harris--Mumford
I second the endorsement of Schaum's Outlines.  

I actually differ from the view that you have to read iconic books to "hang".  If you read any book and do a good job on it (mastery learning) that will drive your comprehension and skills.  Perfection of Schaum's will trump mediocre work through a tougher text any day of the week.  In addition, there are always different tricks or approaches to find if you look at different books.