With the rapid development of microwave photonics technology, high-speed processing and ultra-weak signal detection capability have become the main bottlenecks in many applications. Thanks to the ultra-weak signal detection capability and the extremely low timing jitter properties of single-photon detectors, the combination of single-photon detection and classical microwave photonics technology may provide a solution to break the above bottlenecks. In this paper, we first report a novel concept of single-photon microwave photonics (SP-MWP), a SP-MWP signal processing system with phase shifting and frequency filtering functionalities is demonstrated based on a superconducting nanowire single photon detector (SNSPD) and a successive time-correlated single photon counting (TCSPC) module. Experimental results show that an ultrahigh optical sensitivity down to -100 dBm has been achieved, and the signal processing bandwidth is only limited by the timing jitter of single-photon detectors. In the meantime, the proposed system demonstrates an ultrahigh anti-interference capability, only the signal which is phase locked by the trigger signal in TCSPC can be extracted from the detected signals combining with noise and strong interference. The proposed SP-MWP concept paves a way to a novel interdisciplinary field of microwave photonics and quantum mechanism, named by quantum microwave photonics.