  The main purpose of this paper is to define representations and a cohomology
of color Hom-Lie algebras and to study some key constructions and properties.
We describe Hartwig-Larsson-Silvestrov Theorem in the case of $\Gamma$-graded
algebras, study one-parameter formal deformations, discuss
$\alpha^{k}$-generalized derivation and provide examples.
