Learning roadmap request: compiling a "Mathematics Stack Exchange Undergraduate Bibliography" [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.
 A: 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 

