We describe the advances that we, and others, have reported during the last years in the area of programmable microwave photonic processors. Following a brief historical sketch, we provide a detailed account of the salient theoretical and experimental results recently reported on waveguide mesh optical core processors. The incorporation of a waveguide mesh optical core into the general microwave photonics programmable processor architecture is then addressed. We illustrate through different examples how this processor can be programmed to enable the most important functionalities required in microwave photonics.