Abstract: Let $T=\left(\begin{array}{ll} A & 0 \\ U & B \end{array}\right)$ be a formal triangular matrix ring, where A and B are rings and U is a (B,A)-bimodule. Characterizations of Gorenstein weak injective left T-modules and Gorenstein weak flat right T-modules over the formal triangular matrix ring T are given. Moreover, the weak injective dimensions of left T-modules and weak flat dimensions of right T-modules, Gorenstein weak injective dimensions of left T-modules and Gorenstein weak flat dimensions of right T-modules are discussed.

Key words: formal triangular matrix ring, Gorenstein weak injective module, Gorenstein weak flat module, weak injective module, weak flat module