Insects and hummingbirds remain unmatched in their aerodynamic ability to hover in place in addition to other acrobatic feats such as flying backward and sideways by exploiting flapping-wing motion. Although this remarkable ability is key to making small-scale aircraft, flapping-hovering behavior has been difficult to reproduce artificially because of the challenging stability, power, and aeroelastic phenomena involved. Recent interest in small-scale unmanned air vehicles, especially those capable of hovering like insects and hummingbirds, is driven by many potential applications. A number of flapping machines have been developed, but only two are capable of untethered hovering flight. A key challenge is to demonstrate a stable untethered flapping-hovering ability at a weight and power approximating that of insects and birds where flapping-hovering flight is observed in nature. Here we demonstrate, for the first time, a passively stable 24-g machine capable of flapping-hovering flight at a Reynolds number similar to insects (Re = 8 times 103). This architecture, particularly the passive stability, may help in the design of insect-sized hovering vehicles as well as shed light on the aeroelastic dynamic principles underlying insect flight.