I will survey recent progress in defining and computing categorical enumerative invariants, analogues of Gromov-Witten invariants defined directly from a cyclic A_infinity category and a choice of splitting of the Hodge filtration on its periodic cyclic homology. A proposed definition of such invariants appeared in 2005 in work of Costello, but the original approach had technical problems that made computations impossible. New results allow us to give an alternate definition of Costello's invariants, where explicit computation is possible -- and indeed we apply our results to B-model calculations for elliptic curves and categories of matrix factorizations. My talk is based on joint work with Junwu Tu and Kevin Costello.