It is a standard fact of life in algebraic geometry that (quasi)coherent sheaves---for example, the sheaf of sections of an algebraic vector bundle---on affine varieties are equivalently encoded by their global sections (as modules over the ring of global functions on the variety), whereas on non-affine varieties one loses most information about a sheaf by passing to its space of global sections. A remarkable property of sheaves that are equipped with an action of the sheaf of differential operators, i.e. D-modules, is that there are projective varieties for which this additional structure allows the global sections to capture all the information about the sheaves. Such varieties are called "D-affine" since D-modules on the variety behave as if the variety were affine. I will explain some recent developments toward understanding D-affineness phenomena for some simple examples of varieties with equivariant structure, i.e. algebraic stacks.