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 non-linear 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.