  We study the properties of the relative derived category
$D_{\mathscr{C}}^{b}$($\mathscr{A}$) of an abelian category $\mathscr{A}$
relative to a full and additive subcategory $\mathscr{C}$. In particular, when
$\mathscr{A}=A{\text -}\mod$ for a finite-dimensional algebra $A$ over a field
and $\mathscr{C}$ is a contravariantly finite subcategory of $A$-$\mod$ which
is admissible and closed under direct summands, the $\mathscr{C}$-singularity
category $D_{\mathscr{C}{\text
sg}}$($\mathscr{A}$)=$D_{\mathscr{C}}^{b}$($\mathscr{A}$)/$K^{b}(\mathscr{C})$
is studied. We give a sufficient condition when this category is triangulated
equivalent to the stable category of the Gorenstein category
$\mathscr{G}(\mathscr{C})$ of $\mathscr{C}$.
