Every complex projective algebraic surface $S$ satisfies the inequality \[9\chi({\cal O}_S)\geq c_1^2\geq 2\chi({\cal O}_S)-6.\] This talk will focus on results (recent and less recent) about the algebraic fundamental group of surfaces of general type with $c_1^2$ “small” with respect to $\chi({\cal O}_S)$. In particular some results obtained in colaboration with R. Pardini and C. Ciliberto will be discussed.