In this correspondence we propose track-before-detect (TBD) strategies for space-time adaptive processing (STAP) radars. As a preliminary step we introduce the target and noise models in discrete-time form. Then, resorting to generalized likelihood ratio test (GLRT)-based and ad hoc procedures we derive detectors for two different scenarios (a point better clarified in the body of the correspondence). The preliminary performance assessment, conducted resorting to Monte Carlo simulation, shows that the proposed procedures might be viable means to implement early detection and track initiation of weak moving targets.