  We define a general class of (multiple) integrals of hypergeometric type
associated with the Jacobi theta functions. These integrals are related to
theta hypergeometric series through the residue calculus. In the one variable
case, we get theta function extensions of the Meijer function. A number of
multiple generalizations of the elliptic beta integral [S2] associated with the
root systems $A_n$ and $C_n$ is described. Some of the $C_n$-examples were
proposed earlier by van Diejen and the author, but other integrals are new. An
example of the biorthogonality relations associated with the elliptic beta
integrals is considered in detail.
