The three-level Trigger and DAQ system of ATLASis designed to be very selective while preserving the full physics potential of the experiment; out of the ~1 GHz of proton-proton interactions provided by the LHC at nominal operating conditions, ~200 events/sec are retained. This paper focuses on the muon reconstruction and selection algorithms employed at the last trigger level. One implements an "outside-in" approach; it starts from a reconstruction in the Muon Spectrometer (MS) and performs a backward extrapolation to the interaction point and track combination in the Inner Detector (ID). The other implements an "inside-out" strategy; it starts muon reconstruction from the ID and extrapolates tracks to MS. Algorithm implementations and results on data from real cosmic rays and simulated collisions are described
