The purpose of this paper is to give the optimal turning problem of winged space vehicles at constant altitude an extensive study. The trajectory optimization problem will be formulated into a very general form. The only control variable, i.e., the bank angle, can be solved from a general quadratic equation. Specific topics of interest are the maximum heading increment turn, the maximum range turn, and the turn to a point in minimum time. Particular effort will be devoted to investigate the shapes of the optimal turning trajectories at several different altitudes. (C) 1997 Elsevier Science Ltd.