In this paper, we study completeness of cotorsion pairs in the category of complexes of R-modules. Let (A, B) be a cotorsion pair in R-Mod. It is shown that the cotorsion pair (\widetilde {A}, dg\widetilde {B}) and (\overline {A},\overline {A}^{\perp}) are complete if A is closed under pure submodules and cokernels of pure monomorphisms, where in Gillespie's definitions\widetilde {A} is the class of exact complexes with cycles in A and dg\widetilde {B} is the class of complexes X with components in B such that the complex Hom (A, X) is exact for every complex A\in\widetilde {A}; and\overline {A} is the class of all complexes with components in A. Furthermore, they are perfect. As an application, we get that every complex over a right coherent ring has a Gorenstein flat cover, which generalizes the well-known results on the existence of Gorenstein flat covers.