Active particles that are self-propelled by converting energy into mechanical motion represent an expanding research realm in physics and chemistry. For micrometer-sized particles moving in a liquid ("microswimmers"), most of the basic features have been described by using the model of overdamped active Brownian motion. However, for macroscopic particles or microparticles moving in a gas, inertial effects become relevant such that the dynamics is underdamped. Therefore, recently, active particles with inertia have been described by extending the active Brownian motion model to active Langevin dynamics that include inertia. In this perspective article, recent developments of active particles with inertia ("microflyers," "hoppers," or "runners") are summarized both for single particle properties and for collective effects of many particles. These include inertial delay effects between particle velocity and self-propulsion direction, tuning of the long-time self-diffusion by the moment of inertia, effects of fictitious forces in noninertial frames, and the influence of inertia on motility-induced phase separation. Possible future developments and perspectives are also proposed and discussed.