Bioinspired flapping-wing micro aerial vehicles (FWMAVs) have emerged over the last two decades as a promising new type of robot. Their high thrust-to-weight ratio, versatility, safety, and maneuverability, especially at small scales, could make them more suitable than fixed-wing and multi-rotor vehicles for various applications, especially in cluttered, confined environments and in close proximity to humans, flora, and fauna. Unlike natural flyers, however, most FWMAVs currently have limited take-off and landing capabilities. Natural flyers are able to take off and land effortlessly from a wide variety of surfaces and in complex environments. Mimicking such capabilities on flapping-wing robots would considerably enhance their practical usage. This review presents an overview of take-off and landing techniques for FWMAVs, covering different approaches and mechanism designs, as well as dynamics and control aspects. The special case of perching is also included. As well as discussing solutions investigated for FWMAVs specifically, we also present solutions that have been developed for different types of robots but may be applicable to flapping-wing ones. Different approaches are compared and their suitability for different applications and types of robots is assessed. Moreover, research and technology gaps are identified, and promising future work directions are identified.