Strategies to solve congruence problems

We will review some of the classical strategies to solve congruence problems and discuss the limits to them. We will focus in estimating the number of solutions to \[ f(x,y) \equiv 0 \pmod p \quad 1\le x,y \le M \] where $f$ is some interesting function (polynomial, exponential, etc.). When $M$ is large, the classical approach on character sums/Fourier Analysis allow us to obtain asymptotics for this quantity. Nevertheless, there seems to be a barrier to this method at $M=p^{1/2}$ and new ideas, based on Additive Combinatorics, are required for the case when $M$ is small. We will discuss for some explicit examples the kind of results that can be obtained for very small $M$ and which techniques are exploited.