  Starting with work of Serre, Katz, and Swinnerton-Dyer, theta operators have
played a key role in the study of $p$-adic and $\bmod p$ modular forms and
Galois representations. This paper achieves two main results for theta
operators on automorphic forms on PEL-type Shimura varieties: 1) the analytic
continuation at unramified primes $p$ to the whole Shimura variety of the
$\bmod p$ reduction of $p$-adic Maass--Shimura operators {\it a priori} defined
only over the $\mu$-ordinary locus, and 2) the construction of new $\bmod p$
theta operators that do not arise as the $\bmod p$ reduction of Maass--Shimura
operators. While the main accomplishments of this paper concern the geometry of
Shimura varieties and consequences for differential operators, we conclude with
applications to Galois representations. Our approach involves a careful
analysis of the behavior of Shimura varieties and enables us to obtain more
general results than allowed by prior techniques, including for arbitrary
signature, vector weights, and unramified primes in CM fields of arbitrary
degree.
