Nontopological solitons are stable field configurations which may be formed in a primordial phase transition. We study their cosmic evolution and examine the possibility that such objects could contribute significantly to the energy density of the Universe. As the Universe cools, initially all but the largest lumps evaporate into free particles; those which survive may subsequently enter a brief accretion phase before they "freeze out" at a final size. Although the minimum critical charges which survive depend on particle masses and couplings, we develop an analysis which applies to a wide class of models. In most cases, solitons of moderate size survive the evaporation process only if there is a significant charge asymmetry or if they form at a temperature well below their binding energy per charge.