Parole chiave

**Abstract** 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 $p_g =q=2$, 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 $p_g =q=2$, it turns out that the self-intersection $K_S^2$ of the canonical divisor $K_S$ is $4 \leqslant K_S^2 \leqslant 9$. We will focus on the cases $K^2 _S =5,6$, describing some constructions (endowed with explicit and global equations) developed in a joint work with Fabrizio Catanese.