  This mostly expository article explores recent developments in the relations
between the three objects in the title from an algebro-combinatorial
perspective.
  We prove a formula for Whittaker functions of a real semisimple group as an
integral over a geometric crystal in the sense of Berenstein-Kazhdan. We
explain the connections of this formula to the program of mirror symmetry of
flag varieties developed by Givental and Rietsch; in particular, the integral
formula proves the equivariant version of Rietsch's mirror symmetry conjecture.
We also explain the idea that Whittaker functions should be thought of as
geometric analogues of irreducible characters of finite-dimensional
representations.
