STARE is a real-time SofTwAre REceiver for positioning and monitoring of the fifth-generation (5G) new radio (NR) and long-term evolution (LTE) downlink signals. The real-time operation is achieved by interfacing directly with the software-defined radio (SDR), therefore avoiding the requirement to store the captured signal on a drive and allowing to process signals continuously over arbitrarily long periods. STARE supports multi-channel SDRs and parallel execution of an arbitrary number of tracking channels, which independently acquire and track the desired signals. The design of the tracking stage follows a closed-loop architecture providing a continuous estimation of the delay, Doppler, phase, and signal-to-noise ratio (SNR) of LTE and 5G NR downlink signals. The receiver also supports a post-processing mode, where the signal capture files can be processed without the SDR. STARE is developed in C++ and Python.