Koszul algebras and their syzygies

These are the notes of the lectures of the author at the 2013 CIME/CIRM summer school on Combinatorial Algebraic Geometry. Koszul algebras, introduced by Priddy, are positively graded K-algebras R whose residue field K has a linear free resolution as an R-module. The first part of the notes is devoted to the introduction of Koszul algebras and their characterization in terms of Castelnuovo-Mumford regularity. In the second part we discuss recernt results on the syzygies of Koszul algebras. Finally in the last part we discuss the Koszul property of Veronese algebras and of algebras associated with collections of hyperspaces.