  Khovanov homology is a recently introduced invariant of oriented links in
$\mathbb{R}^3$. It categorifies the Jones polynomial in the sense that the
(graded) Euler characteristic of the Khovanov homology is a version of the
Jones polynomial for links. In this paper we study torsion of the Khovanov
homology. Based on our calculations, we formulate several conjectures about the
torsion and prove weaker versions of the first two of them. In particular, we
prove that all non-split alternating links have their integer Khovanov homology
almost determined by the Jones polynomial and signature. The only remaining
indeterminacy is that one cannot distinguish between $\mathbb{Z}_{2^k}$ factors
in the canonical decomposition of the Khovanov homology groups for different
values of $k$.
