  All commonly considered dark matter scenarios are based on hypothetical
particles with small but non-zero thermal velocities and tiny interaction
cross-sections. A generic consequence of these attributes is the suppression of
small-scale matter perturbations either due to free-streaming or due to
interactions with the primordial plasma. The suppression scale can vary over
many orders of magnitude depending on particle candidate and production
mechanism in the early Universe. While nonlinear structure formation has been
explored in great detail well above the suppression scale, the range around
suppressed perturbations is still poorly understood. In this paper we study
structure formation in the regime of suppressed perturbations using both
analytical techniques and numerical simulations. We develop simple and
theoretically motivated recipes for the halo mass function, the expected number
of satellites, and the halo concentrations, which are designed to work for
power spectra with suppression at arbitrary scale and of arbitrary shape. As
case studies, we explore warm and mixed dark matter scenarios where effects are
most distinctive. Additionally, we examine the standard dark matter scenario
based on weakly interacting massive particles (WIMP) and compare it to pure
cold dark matter with zero primordial temperature. We find that our
analytically motivated recipes are in good agreement with simulations for all
investigated dark matter scenarios, and we therefore conclude that they can be
used for generic cases with arbitrarily suppressed small-scale perturbations.
