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Let A and B be two rings, M be a left B, right A bimodule and be the formal triangular matrix ring. It is known that the category of right T-modules is equivalent to the category Ω of triples (X, Y) f, where X is a right A-module, Y is a right B-module and is a right A-homomorphism. Using this alternative description of right T-modules, in the first part of this paper, we study the vanishing of the extension functor ‘Ext’over T. To this end, we first describe explicitly the structure of (right) T-modules of finite projective (respectively, injective) dimension. Using these results, we shall characterize respectively modules in Auslander's G-class, Gorenstein injective modules, cotorsion modules and tilting and cotilting modules over T. As another application we investigate the structure of the flat covers of right T-modules.