  We study $\mathscr{D}$-elliptic sheaves in terms of their associated modules,
which we call Drinfeld-Stuhler modules. We prove some basic results about
Drinfeld-Stuhler modules and their endomorphism rings, and then examine the
existence and properties of Drinfeld-Stuhler modules with large endomorphism
algebras, which are analogous to CM and supersingular Drinfeld modules.
Finally, we examine the fields of moduli of Drinfeld-Stuhler modules.
