Binary systems showing both apsidal motion and light travel time (LTT) effects which cause orbital period changes in close binaries were studied. 15 triple systems showing apsidal motion were found by searching the literature, and a table including the important parameters of these systems was constructed. Six of the systems given in this table were selected and observed photometrically. Existence of both apsidal motion and LTT effects in all selected systems was investigated by means of the analysis of their eclipse times. The mean observed internal structure constants, log = k(2,obs), and contributions to the apsidal motion from the theory of General Relativity and the third/fourth bodies were calculated. The masses of the third/fourth bodies and some characteristics of their orbits were also calculated.