In this paper, we study the exact and approximate controllability of a class of semilinear integro-differential control systems involving impulses, delays, and nonlocal conditions. In contrast to standard approaches, where the corresponding linear dynamics are transformed into an ordinary differential equation and memory effects are incorporated as perturbations, we preserve the original Volterra integro-differential structure of the system. Our approach starts with a detailed analysis of the controllability properties of the associated linear integro-differential system. Based on this characterization, we then address the semilinear problem. Under appropriate assumptions on the nonlinear contributions, including those induced by delay-dependent terms, impulsive effects, and nonlocal conditions, we establish exact controllability by employing Rothe's fixed-point theorem and a suitable subadditivity framework. Furthermore, by weakening the conditions imposed on the nonlinear terms, we obtain approximate controllability results through an alternative technique inspired by A.E. Bashirov et al., which allows us to avoid the use of fixed-point arguments.