In some mammalian genes, paternally and maternally derived alleles are expressed differently: this phenomenon is called genomic imprinting. Several-explanations have been proposed for the observed patterns of genomic imprinting, but the most successful explanation is the genetic conflict hypothesis--natural selection operating on the gene expression produces the parental origin-dependent gene expression--because the paternally derived allele tends to be less related to the siblings of the same mother than the maternal allele and hence the paternal allele should evolve to be more aggressive in obtaining maternal resources. The successes and failures of this argument have been examined in explaining the observed patterns of genomic imprinting in mammals. After a brief summary of the observations with some examples, a quantitative genetic model describing the evolution of the cis-regulating element of a gene affecting the maternal resource acquisition was presented. The model supports the verbal argument that the growth enhancer should evolve to show imprinting with the paternal allele expressed and the maternal allele inactive, whereas a growth suppressor gene tends to have an inactive paternal allele and an active maternal allele. There are four major problems of the genetic conflict hypothesis. (1) Some genes affect embryonic growth but are not imprinted (e.g., Igf1), which can be explained by considering recessive, deleterious mutations on the coding regions, (2) A gene exists that shows the pattern that is a perfect reversal (Mash2), which is needed for placental growth, and yet has an active maternal allele and an inactive paternal allele. This can be explained if the overproduction of this gene causes dose-sensitive abortion to occur in early gestation. (3) Paternal disomies are sometimes smaller than normal embryos. This is a likely outcome of evolution if imprinted genes control the allocation between placenta and embryo by modifying the cell developmental fate. (4) Genes on X chromosomes do not follow the predictions of the genetic conflict hypothesis. For genes on X chromosomes, two additional forces of natural selection (sex differentiation and dosage compensation) cause genomic imprinting, possibly in the opposite direction. Available evidence suggests that these processes are stronger than the natural selection caused by female multiple mating. Finally, the same formalism of evolution can handle an alternative nonconflict hypothesis: genomic imprinting might have evolved because it reduces the risk of the spontaneous development of parthenogenetic embryo, causing a serious threat to the life of the mother (ovarian time bomb hypothesis). This hypothesis can also explain major patterns of genomic imprinting. In conclusion, the genetic conflict hypothesis is very successful in explaining the observed patterns of imprinting for autosomal genes and probably is the most likely evolutionary explanation for them. However, for genes on X chromosomes, other processes of natural selection are more important. Considering that a nonconflict hypothesis can also explain the patterns in principle, we need a quantitative estimate of various parameters, such as the rate of dose-dependent abortion, the degree of female promiscuity, and the rate of spontaneous development of the parthenogenetic embryo, in order to make judgments on the relative importance of different forces of natural selection to form genomic imprinting.