If one assumes that a population of elements is made up of several subgroups, each subgroup with its own underlying distribution, and the several subgroups mixed together according to certain proportions, there would be an instance of a mixture of distributions i.e., the underlying distribution for the entire population would be a mixture of the distributions for each subgroup. A study is made of the more recent developments in the theory of mixtures of distributions. The problem of identifiability in mixtures is considered in some detail. The special cases of linear mixtures and the distribution of sums of independent random variables are also considered. Finally, the problems encountered in estimation of parameters in mixtures are discussed.