In this paper the changes in number of the various genotypes in a population is formulated as a multidimensional birth and death process. Two classes of populations are considered, namely, bisexual diploid population and asexual haploid populations. Various forms of the birth functions for the two classes of populations are deduced from biological considerations. Four multi-dimensional generalizations of classical one-dimensional cases are presented, and the generalized linear birth and death process is found in some two-and three-dimensional cases, and the mean vector of the process for an arbitrary finite dimension and arbitrary parameters is also studied.