Generation, processing, and sensing of microwave signals using photonic technologies offer immunity from electromagnetic interference, allow high-speed operation over a wide frequency range, and reduce the overall system weight. Recent developments in integrated photonic technologies further increase the attractiveness of photonics-based RF systems. Stimulated Brillouin scattering (SBS), due to its inherently narrow bandwidth, wavelength transparent nature, low threshold, and bandwidth reconfigurability, has emerged as a potential contender for microwave photonic generation and signal processing. In this article, we present a review of recent developments in the area of SBS-based microwave photonics. We discuss different SBS-based techniques for realizing some of the key microwave photonic signal processing functionalities such as radio frequency generation, filtering, and true time delay. Implementation of these tasks using fiber and on-chip platforms is presented along with the effect of system parameters such as dispersion on their performance. Novel architectures to mitigate these impairments are discussed.