Finite mixture models are useful when modeling heterogeneous data sets. It is mainly applicable for the reliability analysis of the systems with multiple components. Finite mixture models combine a finitely many subpopulations, when the subpopulations have infinitely many elements. The stochastic comparison of two finite mixtures is an interesting topic which could be useful to compare two systems with multiple components, heterogeneous in nature. This thesis focuses on the study of various stochastic comparisons of finite mixture random variables. The main goal of the thesis is to establish several ordering results between two finite mixture random variables in terms of the usual stochastic order, hazard rate order, reversed hazard rate order, likelihood ratio order, ageing faster order in terms of reversed hazard rate, dispersive order, star order, Lorenz order, and right-spread order. When random variables of the subpopulations of the mixture models are chosen from general and specific models, such as the general parametric, location- scale, exponentiated location-scale, generalized Weibull, and inverted-Kumaraswamy models, several sufficient conditions have been established to get the comparison results between two finite mixture random observations. Furthermore, the concept of multiple-outlier model has been also introduced for the study of stochastic comparison between two mixture random variables. In addition, multiple numerical examples and counterexamples are shown to demonstrate the efficacy of the established theoretical findings. Finally, a simulation study is conducted to validate some of the theoretical results established in the thesis and presents some concluding remarks with possible real world applications of the established theoretical findings.