  This paper completes the construction of $p$-adic $L$-functions for unitary
groups. More precisely, in 2006, the last three named authors proposed an
approach to constructing such $p$-adic $L$-functions (Part I). Building on more
recent results, including the first named author's construction of Eisenstein
measures and $p$-adic differential operators, Part II of the present paper
provides the calculations of local $\zeta$-integrals occurring in the Euler
product (including at $p$). Part III of the present paper develops the
formalism needed to pair Eisenstein measures with Hida families in the setting
of the doubling method.
