  The \emph{Barnes $\zeta$-function} is \[
  \zeta_n (z, x; \a) := \sum_{\m \in \Z_{\ge 0}^n} \frac{1}{\left(x + m_1 a_1 +
\dots + m_n a_n \right)^z} \] defined for $\Re(x) > 0$ and $\Re(z) > n$ and
continued meromorphically to $\C$. Specialized at negative integers $-k$, the
Barnes $\zeta$-function gives \[
  \zeta_n (-k, x; \a) = \frac{(-1)^n k!}{(k+n)!} \, B_{k+n} (x; \a) \] where
$B_k(x; \a)$ is a \emph{Bernoulli--Barnes polynomial}, which can be also
defined through a generating function that has a slightly more general form
than that for Bernoulli polynomials. Specializing $B_k(0; \a)$ gives the
\emph{Bernoulli--Barnes numbers}. We exhibit relations among Barnes
$\zeta$-functions, Bernoulli--Barnes numbers and polynomials, which generalize
various identities of Agoh, Apostol, Dilcher, and Euler.
