  These lectures discuss recent advances on syzygies on algebraic curves,
especially concerning the Green, the Prym-Green and the Green-Lazarsfeld Secant
Conjectures. The methods used are largely geometric and variational, with a
special emphasis on examples and explicit calculations. The notes are based on
series of lectures given in Daejeon (March 2013), Rome (November-December 2015)
and Guanajuato (February 2016).
