  Let $G$ be a connected reductive group over $\CC$ and let $G^{\vee}$ be the
Langlands dual group. Crystals for $G^{\vee}$ were introduced by Kashiwara as
certain ``combinatorial skeletons'' of finite-dimensional representations of
$G^{\vee}$. For every dominant integral weight of $G^{\vee}$ Kashiwara
constructed a canonical crystal. Other (independent) constructions of those
crystals were given by Lusztig and Littelmann. It was also shown by Kashiwara
and Joseph that the above family of crystals is unique if certain reasonable
conditions are imposed.
  The purpose of this paper is to give another (rather simple) construction of
these crystals using the geometry of the affine Grassmannian
$\calG_G=G(\calK)/G(\calO)$ of the group $G$, where $\calK=\CC((t))$ is the
field of Laurent power series and $\calO=\CC[[t]]$ is the ring of Taylor
series. We check that the crystals we construct satisfy the conditions of the
uniqueness theorem mentioned above, which shows that our crystals coincide with
those constructed in {\it loc. cit}. It would be interesting to find these
isomorphisms directly (cf., however, \cite{Lus3}).
