Let $\mathcal{V}$, $\mathcal{W}$, $\mathcal{Y}$, $\mathcal{X}$ be four classes of left $R$-modules. The notion of $(\mathcal{V, W, Y, X})$-Gorenstein $R$-complexes is introduced, and it is shown that under certain mild technical assumptions on $\mathcal{V}$, $\mathcal{W}$, $\mathcal{Y}$, $\mathcal{X}$, an $R$-complex $\bm{M}$ is $(\mathcal{V, W, Y, X})$-Gorenstein if and only if the module in each degree of $\bm{M}$ is $(\mathcal{V, W, Y, X})$-Gorenstein and the total Hom complexs Hom$_R(\bm{Y},\bm{M})$, Hom$_R(\bm{M},\bm{X})$ are exact for any $\bm{Y}\in\widetilde{\mathcal{Y}}$ and any $\bm{X}\in\widetilde{\mathcal{X}}$. Many known results are recovered, and some new cases are also naturally generated.