The repetitively pulsed streamer discharge is a critical enabling factor in many advanced low-temperature plasma applications. However, the streamer discharge exhibits complex instabilities and memory effect phenomena under high-frequency repetitive pulses. Fundamental discharge evolution mechanisms and regulation methods are not thoroughly understood, which significantly affects the application safety and regulation efficiency of discharge properties. In this paper, evolution phenomena and mechanisms of repetitively pulsed streamer discharge are reviewed, strong nonlinearity and progressive evolution features are summarized for repetitively pulsed streamer discharge, different memory effect agents and their influential mechanisms on initiation and propagation of subsequent streamers are discussed, effects of pulse waveform parameters on repetitively pulsed streamer discharge are outlined, and several research challenges are proposed regarding evolution mechanisms of repetitively pulsed streamer discharge, which would be helpful for revealing mechanisms of pulsed discharge plasmas.