Autonomous flight in confined or cluttered environments such as houses or urban canyons requires high manoeuvrability, fast mapping from sensors to actuators and very limited overall system weight. Although flying animals are well capable of solving this problem, roboticists have difficulties at reproducing such capabilities. This paper describes why and how we took inspiration from flying insects to progress toward the goal of developing small UAVs able to dynamically fly in cluttered environments. This endeavour allowed us to demonstrate a 10-gram microflyer capable of fully autonomous operation in an officesized room using fly-inspired vision, inertial and airspeed sensors. This encouraging result is now being ported to outdoor scenarios such as low-altitude flight in urban or mountainous environments. Important is that these autonomous capabilities are achieved without the help of GPS or active range finders, which allows to develop very lightweight autopilots.