  This is a survey paper on recent work on syzygies of algebraic varieties. We
discuss the gonality conjecture on weight-one syzygies of algebraic curves,
syzygies of secant varieties of algebraic curves, syzygies of tangent
developable surfaces and Green's conjecture on syzygies of canonical curves,
and asymptotic syzygies of algebraic varieties. All results considered in this
paper were proven using the geometry of symmetric products of algebraic curves.
