  We prove that limits of multiplicities associated to graded families of
ideals exist under very general conditions. Most of our results hold for
analytically unramified equicharacteristic local rings, with perfect residue
fields. 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 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
filtration on a family of algebraic objects.
