We present an overview of scenarios where the observed Dark Matter (DM) abundance consists of Feebly Interacting Massive Particles (FIMPs), produced nonthermally by the so-called freeze-in mechanism. In contrast to the usual freeze-out scenario, frozen-in FIMP DM interacts very weakly with the particles in the visible sector and never attained thermal equilibrium with the baryon–photon fluid in the early Universe. Instead of being determined by its annihilation strength, the DM abundance depends on the decay and annihilation strengths of particles in equilibrium with the baryon–photon fluid, as well as couplings in the DM sector. This makes frozen-in DM very difficult but not impossible to test. In this review, we present the freeze-in mechanism and its variations considered in the literature (dark freeze-out and reannihilation), compare them to the standard DM freeze-out scenario, discuss several aspects of model building, and pay particular attention to observational properties and general testability of such feebly interacting DM.