Summary: We establish the existence of solutions of a more general class of stochastic integral equations of Volterra type. The main tools used here are the measure of noncompactness and the fixed point theorem of Darbo. The results generalize the results of C. P. Tsokos and W. J. Padgett [“Random integral equations with applications to life sciences and engineering” (New York, 1974)] and D. Szynal and S. Wedrychowicz [J. Appl. Probab. 25, No. 2, 257-267 (1988; Zbl 0643.60048)]. An application to a stochastic model arising in chemotherapy is discussed.