There has been recent interest in Hom-xxx structures, where xxx may be Lie, associative, Nambu, Poisson, alternative, Hopf, to name a few. I'll survey the basics of the hom theory (mainly in the Lie and associative cases) with some history and comments on cohomology and deformations of these structures. Moderator's comment: The Hom nomenclature is an unfortunate historical accident.