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.

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.
