The classification of complex algebraic surfaces

This text grew up from the notes of a graduate course I gave at the University of Roma ``Tor Vergata'' in the academic year 2018--19. The subject is the classification of complex algebraic surfaces following Mori's theory. It includes the $P_{12}$--Theorem, Sarkisov's programme in the surface case and Noether--Castelnuovo's Theorem in its classical version.