  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.
