On the Classification of Surfaces of General Type with pg=q=2

The classification of minimal surfaces of general type is a classical and long-standing research topic. In this context fixing invariants turns out to be fundamental.
In this talk we will review some recent results on the case pg=q=2, which is still widely open in spite of several contributions by many authors over the last two decades. More specifically, given a minimal surface S of general type with pg=q=2, it turns out that the self-intersection K^2 of the canonical divisor K is between 4 and 9. We will focus on the cases K^2=5,6, describing some constructions (endowed with explicit and global equations) developed in a joint work with Fabrizio Catanese.