Every complex projective algebraic surface S satises the inequality
$9\chi(\mathcal{O}_S) \geq c_1^2 \geq 2\chi(\mathcal{O}_S) -6$
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 $\chi(\mathcal{O}_S)$. In particular some recent results (obtained in colaboration with R. Pardini and C. Ciliberto) will be discussed.