  We extend Morita theory to abelian categories by using wide Morita contexts.
Several equivalence results are given for wide Morita contexts between abelian
categories, widely extending equivalence theorems for categories of modules and
comodules due to Kato, Muller and Berbec. In the case of Grothendieck
categories we derive equivalence results by using quotient categories. We apply
the general equivalence results to rings with identity, rings with local units,
graded rings, Doi-Hopf modules and coalgebras.
