Reading Recommendations
The world would be a great place if I could write about everything I knew about, but alas I have a finite amount of time. So in addition to the stuff I have on this website, here's a list of other resources I like.
Please notify me of any broken links, suggestions, etc. by email.
An abridged version of this page for olympiad students can be found here.
Undergraduate Math and Computer Science

MSci Category Theory notes by Tom Leinster. I highly enjoyed these notes; very carefully written and explains intuition. Some minimal knowledge of group theory and linear algebra is used in the examples. See instead the corresponding print book.

Analytic NT notes by AJ Hildebrand. A set of lecture notes for analytic number theory, suitable for selfstudy. A light introduction where you get to prove versions of the Prime Number Theorem and Dirichlet's Theorem.

Algebraic Geometry by Andreas Gathmann. My preferred introduction to algebraic geometry; short but complete. This was the source that finally got me to understand the concept of a ringed space. It doesn't officially cover schemes, but because it covers general varieties as ringed spaces the fullfledged scheme is not much harder.

Manifolds and Differential Forms by Reyer Sjamaar. My preferred introduction to differential geometry; very readable and works with minimal prerequisites. Also, beautifully drawn figures.

Harvard's CS 125: Algorithms and Complexity has delightful lecture and section notes.
Of course, see also Napkin.
Olympiad Resources
Handouts

My own handouts (sorry, couldn't resist linking them again).

Yufei Zhao's site has several excellent handouts, especially in geometry. I consulted many of them when I was coming up with ideas for my geometry textbook. In particular, the Cyclic Quadrilaterals handout is especially worth reading.

Alexander Remorov, in particular the projective geometry handout, which the corresponding chapter in my textbook is based off of.

PoShen Loh, mostly combinatorics. See especially the handouts on the probabilistic method.
Books

Olympiad NT through Challenging Problems, by Justin Stevens, is an introductory olympiad number theory text, at a level somewhat easier than what my own number theory handouts assume.
Contests
Each section is in alphabetical order.
 National olympiads:
 Team selection tests:
 International contests: