The concept of cohomology is one of the most subtle and powerful in modern mathematics. While its application to topology and integrability is immediate (it was probably how cohomology was born in the first place), there are many more fields in which cohomology is at least a very interesting point of view. Group cohomology is a famous one, and for example it helps in studying extensions.
Here are good points about the "philosophy" behind cohomology.
Here are very good, but advanced, ideas on what cohomology "really is".
I would like to ask something a little different:
What are the most unexpected applications of cohomology, or of cohomology-related ideas? Why is cohomology useful/important/interesting when applied to such problems?
Bonus point for real-world applications, or at least outside algebra/geometry/theoretical physics.
Update: Oops, looks like there is a very similar question here, with beautiful answers.
Best Answer
Here is a ridiculous application of cohomology: a proof of $$\sum_{j=0}^n {n \choose j} (-1)^j=0.$$
Let $X=(S_1)^n$ be the $n$-dimensional torus. By the Künneth formula, $H^j(X, \mathbf Q)$ has dimension ${n \choose j}$. Therefore, the Euler characteristic of $X$ is
$$\chi(X)=\sum_{j=0}^n (-1)^j \mathrm{dim}_{\mathbf Q}H^j(X, \mathbf Q) = \sum_{j=0}^n {n \choose j} (-1)^j.$$
On the other hand, $X$ is a compact Lie group; let $\sigma$ be an infinitesimal translation $X \to X$. By the Lefschetz fixed point theorem, $\chi(X)$ is equal to the number of fixed points of $\sigma$, i.e., $0$.