Fourier series are useful (and sometimes essential) for solving/understanding many problems involving periodic functions on $\mathbb{R}$ or, equivalently, functions $f$ on $[a,b]$ such that $f(a)=f(b)$. I was going to say almost all problems, but that's probably an exaggeration. Of course it helps if the problem is linear, and the properties of the functions you're considering can be easily expressed in terms of the Fourier coefficients -- but even then these restrictions are not always essential.
My favourite one: proving the Isoperimetric Inequality, that the circle has the largest area of all piecewise $C^1$ curves with given perimeter;
The functional equation for the Riemann Zeta Function $\zeta$: one proof involves the Fourier expansion of the sawtooth function $x - [x]$, which I think I saw in E. C. Titchmarsh's old book The Theory of the Riemann Zeta Function (although I'm sure many other books will give it also).