LetV be a system of weights on a completely regular Hausdorff spaceX and letB(E) be the topological vector space of all continuous linear operators on a general topological vector spaceE. LetCV0(X, E) andCVb(X, E) be the weighted spaces of vector-valued continuous functions (vanishing at infinity or bounded, respectively) which are not
necessarily locally convex. In the present paper, we characterize in this general setting the weighted composition operatorsWπ,ϕ onCV0(X, E) (orCVb(X, E)) induced by the operator-valued mappings π:X→B(E) (or the vector-valued mappings π:X→E, whereE is a topological algebra) and the self-map ϕ ofX. Also, we characterize the mappings π:X→B(E) (or π:x→E) and ϕ:X→X which induce the compact weighted composition operators on these weighted spaces of continuous functions.