  This is the first paper in a series. We develop a general deformation theory
of objects in homotopy and derived categories of DG categories. Namely, for a
DG module $E$ over a DG category we define four deformation functors $\Def
^{\h}(E)$, $\coDef ^{\h}(E)$, $\Def (E)$, $\coDef (E)$. The first two functors
describe the deformations (and co-deformations) of $E$ in the homotopy
category, and the last two - in the derived category. We study their properties
and relations. These functors are defined on the category of artinian (not
necessarily commutative) DG algebras.
