As an emerging topic, photonic-assisted microwave measurements with distinct features such as wide frequency coverage, large instantaneous bandwidth, low frequencydependent loss, and immunity to electromagnetic interference, have been extensively studied recently. In this article, we provide a comprehensive overview of the latest advances in photonic microwave measurements, including microwave spectrum analysis, instantaneous frequency measurement, microwave channelization, Doppler frequency-shift measurement, angle-of-arrival detection, time-frequency analysis, compressive sensing, and phase-noise measurement. A photonic microwave radar, as a functional measurement system, is also reviewed. The performance of the photonic measurement solutions is evaluated and compared with the electronic solutions. Future prospects using photonic integrated circuits and software-defined architectures to further improve the measurement performance are also discussed.