  We start the study of reduced complex projective plane curves, whose Jacobian
syzygy module has 3 generators. Among these curves one finds the nearly free
curves introduced by the authors, and the plus-one generated line arrangements
introduced by Takuro Abe. All the Thom-Sebastiani type plane curves, and more
generally, any curve whose global Tjurina number is equal to a lower bound
given by A. du Plessis and C.T.C. Wall, are 3-syzygy curves. Rational plane
curves which are nearly cuspidal, i.e. which have only cusps except one
singularity with two branches, are also related to this class of curves.
