  We find simple necessary and sufficient conditions on a local ring $R$ of
dimension $d$ for the limit $$
\lim_{i\rightarrow\infty}\frac{\ell_R(R/I_n)}{n^d} $$ to exist whenever
$\{I_n\}$ is a graded family of $m_R$-primary ideals, and give a number of
applications. We also give simple necessary and sufficient conditions on
projective schemes over a field $k$ for asymptotic limits of the growth of all
graded linear series of a fixed Kodaira-Iitaka dimension to exist.
