The authors present results for the measured thermopower of a number of YBa₂Cu₃O_{7- delta} single crystals. One of the crystals was untwinned and allowed to measure the thermopower in the a and b crystallographic directions and discuss the contributions to the thermopower from the chains and planes of Cu-O. The thermopower near T_c is fitted to recent models of superconducting fluctuation effects in the thermopower and it is found that the a axis coherence length is almost five times the b axis coherence length. A sharp peak near T_c is nearly always observed and they argue that while this may be a superconducting fluctuation effect it may also be the result of a combination of the a axis and b axis thermopowers which are of opposite sign. The thermopower in the b direction is nearly temperature independent well above T_c, which is consistent with conduction in a narrow band. The a axis thermopower, due essentially to the Cu-O₂planes, is metallic-like and similar to results on other high-T_c systems.