Space-time adaptive processing (STAP) is a well-known and effective method to detect targets, obscured by interference, from airborne radars that works by coherently combining signals from a phased antenna array (spatial domain) with multiple radar pulses (temporal domain). As widely demonstrated, optimum STAP, in the sense of maximizing the output signal to interference plus noise ratio (SINR), is a coherent, linear, transversal filter (i.e., tapped delay line), that can be synthesized by a complex-valued weight vector. This paper extends previous work that focused on adaptive spatial-only nulling; it derives the optimum phase-only STAP, namely, the optimal weight vector that maximizes the SINR subject to the constraint it belongs to the $N$ -torus of phase-only complex vectors, where $N$ is the number of spatio-temporal degrees of freedom. Because this problem does not admit a closed-form solution, it is solved numerically using the phase-only conjugate gradient method (CGM). The effectiveness of phase-only STAP is demonstrated using both SINR values and receiving beampattern shape, comparing it with the optimum fully-adapted STAP and the nonadapted beam former responses as well as other possible counterparts. Additionally, several analyses of practical utility also demonstrate the benefits provided by phase-only STAP.