  In this paper, we introduce the notion of a (regular) Hom-Lie group. We
associate a Hom-Lie algebra to a Hom-Lie group and show that every regular
Hom-Lie algebra is integrable. Then, we define a Hom-exponential (Hexp) map
from the Hom-Lie algebra of a Hom-Lie group to the Hom-Lie group and discuss
the universality of this Hexp map. We also describe a Hom-Lie group action on a
smooth manifold. Subsequently, we give the notion of an adjoint representation
of a Hom-Lie group on its Hom-Lie algebra. At last, we integrate the Hom-Lie
algebra $(\mathfrak{gl}(V),[\cdot,\cdot],\mathsf{Ad})$, and the derivation
Hom-Lie algebra of a Hom-Lie algebra.
