Microwave photonics has recently come to the forefront as a valuable approach to generating, processing, and measuring signals in high-performance domains such as communication, radar, and timing systems. Recent studies have introduced a range of photonics-based phase-noise analyzers (PNAs) that utilize a variety of architectures, including phase detection, frequency discrimination, and hybrid mechanisms that combine optical with electronic processing. This review delves into the microwave photonics methodologies developed with the specific purpose of measuring phase noise, by exploring their fundamental principles, system design frameworks, and performance indicators. Through the integration of insights garnered from recent publications, our objective is to deliver a comprehensive understanding of the strengths and limitations associated with PNAs and to pinpoint new and promising areas for advancing research in the field of oscillator metrology.