Abstract. Let R be a ring and M be a left R-module. M is called a cofinitely generalized (weak) δ-supplemented module or briefly a δ-CGS-module (δ-CGWS-module) if every cofinite submodule of M has a generalized (weak) δ-supplement in M . In this paper, we give various properties of these modules. It is shown that (1) The class of cofinitely generalized (weak) δ-supplemented modules are closed under taking homomorphic images, arbitrary sums, generalized δ-covers and closed under extensions. (2) M is a generalized cofinitely δ-semiperfect module if and only if M is a cofinitely generalized δ-supplemented by generalized δ-supplements which have generalized projective δ-covers.