The paper is devoted to the study of the nonlinear Volterra integral equations
in the Banach space consisting of all real functions defined, bounded and continuous on . Using the measures of noncompactness and the fixed point theorem, conditions are given when equation (1) has at least one globally attractive solution and when equation (2) has at least one asymptotically stable solution. The last statement for the special case of equation (2) with improves the corresponding result of J. Banas and B. Rzepka [J. Math. Anal. Appl. 284, No. 1, 165–173 (2003; Zbl 1029.45003)]. Three examples are given.