In 1935, Erdős and Szekeres proved that, for every positive integer k, every sufficiently large point set contains a "k-gon", that is, a subset of k points which is the vertex set of a convex polygon. Their theorem is a classical result in both, combinatorial geometry and Ramsey theory, and motivated a lot of further research including numerous modifications and extensions of the theorem. In this talk we discuss some results and methods that played an essential role in the study of k-gons and the variant of "k-holes".