  Let $A$ and $B$ be rings, $U$ a $(B, A)$-bimodule and
$T=\left(\begin{smallmatrix} A & 0 \\ U & B \\\end{smallmatrix}\right)$ be the
triangular matrix ring. In this paper, we characterize the Gorenstein
homological dimensions of modules over $T$, and discuss when a left $T$-module
is strongly Gorenstein projective or strongly Gorenstein injective module.
