A classification of hypotheses on the advantage of amphimixis over apomixis is presented. According to "Immediate Benefit" hypotheses, amphimixis is advantageous regardless of reciprocal gene exchange, because either it directly increases fitness of the progeny, reduces the deleterious mutation rate, or makes selection more efficient. In contrast, "Variation and Selection" hypotheses attribute the advantage of amphimixis to the reciprocal gene exchange that alters genetic variability and response to selection among the progeny. Most such hypotheses assume that amphimixis increases variability and efficiency of selection, but some claim that amphimixis decreases response to selection. Variation and Selection hypotheses require that some factor, either random drift or epistatic selection, makes distributions of different alleles nonindependent, while another factor, either changes of the genotype fitnesses or deleterious mutations, makes overrepresented genotypes non-optimal. Numerous Variation and Selection hypotheses, dealing with either unstructured or spatially structured populations, are reviewed. Two of them seem most plausible: better responsiveness of the amphimictic population to widely fluctuating selection, and lower mutation load in the amphimictic population under synergistic selection against deleterious mutations. In both cases the large advantage of amphimixis requires rather stringent conditions, which could be falsified by careful experiment. Further progress in understanding the evolution of amphimixis will depend mostly on such experimental work.