  A twisted generalization of Lie-Yamaguti algebras, called Hom-Lie-Yamaguti
algebras, is defined. Hom-Lie-Yamaguti algebras generalize Hom-Lie triple
systems (and susequently ternary Hom-Nambu algebras) and Hom-Lie algebras in
the same way as Lie-Yamaguti algebras generalize Lie triple systems and Lie
algebras. It is shown that the category of Hom-Lie-Yamaguti algebras is closed
under twisting by self-morphisms. Constructions of Hom-Lie-Yamaguti algebras
from classical Lie-Yamaguti algebras and Malcev algebras are given. It is
observed that, when the ternary operation of a Hom-Lie-Yamaguti algebra
expresses through its binary one in a specific way, then such a
Hom-Lie-Yamaguti algebra is a Hom-Malcev algebra.
