In this talk, we shall define $\Gamma$-contractions, which were introduced by Jim Agler and Nicholas Young. We shall construct an
 explicit $\Gamma$-isometric dilation of a $\Gamma$-contraction and produce a genuine functional model. A crucial operator equation 
 has to be solved for constructing such a dilation. We shall show how the existence of such a solution characterizes a $\Gamma$-con-
 traction. This solution, which is unique, also provides a complete unitary invariant.