  In this paper we develop a new method to study Iwasawa theory and Eisenstein
families for unitary groups $\mathrm{U}(r,s)$ of general signature over a
totally real field $F$. As a consequence we prove that for a motive
corresponding to a regular algebraic cuspidal automorphic representation $\pi$
on $\mathrm{U}(r,s)_{/F}$ which is ordinary at $p$, twisted by a Hecke
character, if its Selmer group has rank $0$, then the corresponding central
$L$-value is nonzero. This generalizes a result of Skinner-Urban in their ICM
2006 report in the special case when $F=\mathbb{Q}$ and the motive is conjugate
self-dual. Along the way we also obtain $p$-adic functional equations for the
corresponding $p$-adic $L$-functions and $p$-adic families of Klingen
Eisenstein series. Our method does not involve computing Fourier-Jacobi
coefficients (as opposed to previous work which only work in low rank cases,
e.g. $\mathrm{U}(1,1)$, $\mathrm{U}(2,0)$ and $\mathrm{U}(1,0)$) whose
automorphic interpretation is unclear in general.
