  Let $R$ be a ring with identity and $\C(R)$ denote the category of complexes
of $R$-modules. In this paper we study the homotopy categories arising from
projective (resp. injective) complexes as well as Gorenstein projective (resp.
Gorenstein injective) modules. We show that the homotopy category of projective
complexes over $R$, denoted $\KPC$, is always well generated and is compactly
generated provided $\KPR$ is so. Based on this result, it will be proved that
the class of Gorenstein projective complexes is precovering, whenever $R$ is a
commutative noetherian ring of finite Krull dimension. Furthermore, it turns
out that over such rings the inclusion functor $\iota :
\K(\RGPrj)\hookrightarrow \KR$ has a right adjoint $\iota_{\rho}$, where
$\K(\RGPrj)$ is the homotopy category of Gorenstein projective $R$ modules.
Similar, or rather dual, results for the injective (resp. Gorenstein injective)
complexes will be provided. If $R$ has a dualising complex, a
triangle-equivalence between homotopy categories of projective and of injective
complexes will be provided. As an application, we obtain an equivalence between
the triangulated categories $\K(\RGPrj)$ and $\K(\RGInj)$, that restricts to an
equivalence between $\KPR$ and $\KIR$, whenever $R$ is commutative, noetherian
and admits a dualising complex.
