  In this paper we introduce geometric crystals and unipotent crystals which
are algebro-geometric analogues of Kashiwara's crystal bases. Given a reductive
group G, let I be the set of vertices of the Dynkin diagram of G and T be the
maximal torus of G. The structure of a geometric G-crystal on an algebraic
variety X consists of a rational morphism \gamma:X-->T and a compatible family
e_i:G_m\times X-->X, i\in I of rational actions of the multiplicative group G_m
satisfying certain braid-like relations. Such a structure induces a rational
action of W on X. Quite surprisingly, many interesting rational actions of the
group W come from geometric crystals. Also all the known examples of the action
of W which appear in the construction of gamma-functions for the
representations of ^LG in the recent work by A. Braverman and D. Kazhdan come
from geometric crystals. There are many examples of positive geometric crystals
on (G_m)^l, i.e., those geometric crystals for which the actions e_i and the
morphism \gamma are given by positive rational expressions. To each positive
geometric crystal X we associate a Kashiwara's crystal corresponding to the
Langlands dual group ^LG. An emergence of ^LG in the "crystal world" was
observed earlier by G. Lusztig. Another application of geometric crystals is a
construction of trivialization which is an W-equivariant isomorhism
X-->\gamma^{-1}(e) \times T for any geometric SL_n-crystal. Unipotent crystals
are geometric analogues of normal Kashiwara crystals. They form a strict
monoidal category. To any unipotent crystal built on a variety X we associate a
certain gometric crystal.
