We prove that limits of multiplicities associated to graded families of ideals exist under very general conditions. Most of our results hold for reduced excellent equicharacteristic local rings, with perfect residue elds. We give a number of applications, including a "volume = multiplicity" formula, generalizing the formula of Lazarsfeld and Mustata, and a proof that the epsilon multiplicity of Ulrich and Validashti exists as a limit for ideals in rather general rings, including analytic local domains. We prove a generalization of this to generalized symbolic powers of ideals, proposed by Herzog, Puthenpurakal and Verma. We also prove an asymptotic "additivity formula" for limits of multiplicities, and a formula on limiting growth of valuations, which answers a question posed by the author, Kia
Dalili and Olga Kashcheyeva. Our proofs are inspired by a philosophy of Okounkov, for computing limits of multiplicities as the volume of a slice of an appropriate cone generated by a semigroup determined by an appropriate ltration on a family of algebraic objects.