  This is a survey paper, based on lectures given at the Workshop on
"Structured ring spectra and their applications" which took place January
21-25, 2002, at the University of Glasgow.
  The term `Morita theory' is usually used for results concerning equivalences
of various kinds of module categories. We focus on the covariant form of Morita
theory, so the basic question is: When do two `rings' have `equivalent' module
categories ?
  We discuss this question in different contexts and illustrate it by examples:
  (Classical) When are the module categories of two rings equivalent as
categories ?
  (Derived) When are the derived categories of two rings equivalent as
triangulated categories ?
  (Homotopical) When are the module categories of two ring spectra Quillen
equivalent as model categories ?
  There is always a related question, which is in a sense more general: What
characterizes the category of modules over a `ring' ?
  The answer is, mutatis mutandis, always the same: modules over a `ring' are
characterized by the existence of a `small generator', which plays the role of
the free module of rank one. The precise meaning of `small generator' depends
on the context, be it an abelian category, a derived category or a stable model
category.
