Semilinear, infinite-dimensional control systems with lumped and distributed delays are considered. It is generally assumed that the control system has both a linear part which contains the lumped and distributed delays and a nonlinear part without delays. Using the Leray-Schauder degree theory, a sufficient condition for approximate controllability in a given finite-time interval is formulated and proved. Moreover, the regularity of the solution of the abstract retarded differential equation is discussed. The relationships between the attainable sets for the semilinear control system and the corresponding linear control system are investigated. As an illustrative example, approximate controllability of the semilinear heat equation is considered. It should be pointed out that the results given in the paper extend the controllability considerations given in the paper [

*K. Naito*, J. Optimization Theory Appl. 60, 57-65 (1989;

Zbl 0632.93007)].