TITLE: Learning roadmap request: compiling a "Mathematics Stack Exchange Undergraduate Bibliography"
QUESTION [17 upvotes]: [Book recommendation] questions are quite popular on this website, which is, at least for me, one of the best places to get useful and insightful suggestions because it gathers a great number of experts in different areas of mathematics. 
In my opinion, it would be very beneficial for the organization of the site (and for the many undergraduate students searching for guidance when facing a new course or looking for a good learning roadmap -- like myself) to make one thread that collects a big "Mathematics Stack Exchange Undergraduate Mathematics Bibliography" divided by categories just like the nice one proposed here, but surely more comprehensive (given the much larger number of contributors).
This thread should collect sparse material already available on the website but hard to find among the numerous questions and also new inputs. Ideally each entry should be briefly commented with matter-of-fact remarks.

REPLY [7 votes]: Please, add your contributions to this answer (which I made community wiki). In case you wish to add some personal comments about your entries, add a tag like [Your_Username].

Foundations
 Daniel J. Velleman, How
    to Prove It: A Structured Approach 
Problem solving

G. Polya, How to Solve It: A New Aspect of Mathematical Method 

Recreational
Algebra

Roger Godement  , Algebra     (Translation of Cours d' Algebre)
Lang - Algebra
Jacobson - Basic Algebra I, Basic Algebra II
Rotman - Introduction to the Theory of Groups
Dummit and Foote - Algebra
Hungerford - Abstract Algebra
Serre - Linear Representations of Finite Groups
Fulton, Harris - Representation Theory - A First Course
Humphreys - Introduction to Lie Algebras and Representation Theory
Atiyah, MacDonald - Introduction to Commutative Algebra
Eisenbud - Commutative Algebra: with a View Toward Algebraic Geometry 
Weibel - An Introduction to Homological Algebra

Linear algebra

Sheldon Axler, Linear Algebra Done Right.

 Sergei Treil, *Linear Algebra Done Wrong*
     

Roman - Advanced Linear Algebra

Geometry

Mumford - The Red Book of Varieties and Schemes

Point-set topology
Differential geometry
 John Lee, *Introduction to Smooth Manifolds
* 
Number theory
  Andre Weil, *Basic Number Theory* 

Ireland and Rosen - A Classical Introduction to Moder Number Theory
Edmund Landau - Introduction to Number Theory
Burton - Elementary Number Theory

Combinatorics and discrete mathematics

Graham, Knuth, Patashnik, Concrete Mathematics.

Probability and Statistics
Real analysis
 Gerald B. Folland, *Real Analysis: Modern Techniques and Their Applications* 

Shilov and  Gurevich: Integral, Measure and Derivative.

Multivariable calculus

Spivak, Calculus On Manifolds.

Complex analysis

Tristan Needham, Visual Complex Analysis.
Remmert - Introduction to the Theory of Functions.
Remmert - Classical Topics in Function Theory

Differential equations
Functional analysis
 Walter Rudin, *Functional Analysis*
Erwin Kreyszig, *Introductory Functional Analysis with Applications*
Reinhold Meise & Dietmar Vogt: Introduction to Functional Analysis 
Mathematical and theoretical physics

Arnold - Mathematical Methods of Classical Mechanics