Landmarks of subjects of mathematics In order to have a good view of the whole mathematical landscape one might want to know a deep theorem from the main subjects (I think my view is too narrow so I want to extend it).
For example in natural number theory it is good to know quadratic reciprocity and in linear algebra it's good to know the Cayley-Hamilton theorem (to give two examples).
So, what is one (per post) deep and representative theorem of each subject that one can spend a couple of months or so to learn about? (In Combinatorics, Graph theory, Real Analysis, Logic, Differential Geometry, etc.)

Differential Geometry: the Gauss-Bonnet theorem. 
I took a one-semester intro course on differential geometry class and we got to this towards the end of the semester, so I feel that a couple of months is an appropriate time frame for this theorem.
