TITLE: What books must every math undergraduate read?
QUESTION [237 upvotes]: 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.

REPLY [5 votes]: Alex Youcis's and other users suggestions are pretty good but I think some of the the books that they mentioned are a little difficult for average undergraduate students. But I totally agree with most of their suggestions.
Here are some books which I found really nice and helpful for undergraduate level:
Algebra :

Algebra in Action: A Course in Groups, Rings, and Fields by Shahriar Shahriari
Abstract Algebra: An Introduction with Applications by Derek Robinson

Linear Algebra :

Linear Algebra Done Right by Sheldon Axler

Analysis :

Real Mathematical Analysis by Charles C. Pugh
Measure, Integration and Real analysis by Sheldon Axler (More Advanced)

Complex Analysis :

Complex Analysis by Theodore W. Gamelin
A Course in Complex Analysis and Riemann Surfaces by Wilhelm Schlag (More Advanced)

Algebraic Topology :

Basic Topology by M. A. Armstrong
Introduction to Topological Manifolds by John M. Lee

Algebraic Geometry :

An Invitation to Algebraic Geometry by Karen E. Smith & others

Smooth Manifolds :

An Introduction to Manifolds by Loring W. Tu

Elementary Number Theory :

Elementary Number Theory and Its Applications by Kenneth H. Rosen

Elliptic Curves :

Rational Points on Elliptic Curves by Joseph H. Silverman & John T. Tate

Dynamical Systems :

Differential Equations, Dynamical Systems, and an Introduction to Chaos by Morris W. Hirsch, Stephen Smale & Robert L. Devaney
Complex Dynamics by Lennart Carleson & Theodor W. Gamelin
Dynamics in One Complex Variable by John W. Milnor