  Let $X$ be an abelian variety defined over an algebraically closed field $k$.
We consider theta groups associated to \emph{simple semi-homogenous vector
bundles of separable type} on $X$. We determine the structure and
representation theory of these groups. In doing so we relate work of Mumford,
Mukai, and Umemura. We also consider adelic theta groups associated to line
bundles on $X$. After reviewing Mumford's construction of these groups we
determine functorial properties which they enjoy and then realize the
Neron-Severi group of $X$ as a subgroup of the cohomology group $\H^2(\V(X);
k^\times)$.
