  We give a classification, up to Morita equivalence, of 2-blocks of
quasi-simple groups with abelian defect groups. As a consequence, we show that
Donovan's conjecture holds for elementary abelian 2-groups, and that the
entries of the Cartan matrices are bounded in terms of the defect for arbitrary
abelian 2-groups. We also show that a block with defect groups of the form
$C_{2^m} \times C_{2^m}$ for $m \geq 2$ has one of two Morita equivalence types
and hence is Morita equivalent to the Brauer correspondent block of the
normaliser of a defect group. This completes the analysis of the Morita
equivalence types of 2-blocks with abelian defect groups of rank 2, from which
we conclude that Donovan's conjecture holds for such 2-groups. A further
application is the completion of the determination of the number of irreducible
characters in a block with abelian defect groups of order 16. The proof uses
the classification of finite simple groups.
