ATLAS will upgrade its muon spectrometer for high-luminosity LHC runs. The number of trigger layers will increase in both barrel (by adding a new layer of Resistive Plate Chamber - RPC) and endcap (by upgrading the EIL4 Thin Gap Chambers - TGC) regions. To create space for the additional layer of RPCs in small sectors, the present Monitored Drift Tube (MDT) chambers need to be replaced by small-diameter MDT chambers. A TGC tracking trigger will be implemented using segments found by TGC chambers in the endcap middle station and by the New Small Wheel (NSW) detector in the endcap inner station. MDT data will be used for the first time at the first trigger level to improve the trigger muon momentum resolution. In addition, most electronics will be replaced to handle higher data rates and longer latency expected at high-luminosity LHC runs. All muon spectrometer upgrade activities are on track for an installation starting at the beginning of 2024.