  We present an overview of scenarios where the observed Dark Matter (DM)
abundance consists of Feebly Interacting Massive Particles (FIMPs), produced
non-thermally 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.
