recent results on syzygies of projective varieties concentrate on their asymptotics. in this talk, i will discuss results on the asymptotics of syzygies under group actions. in particular, we study two cases. when the underlying space is the projective space, we give the asymptotic growth of syzygy modules with respect to the general linear group. when the underlying space is a toric variety, we give a sharp asymptotic description of the distribution of torus weights.