In the absence of randomized controlled and natural experiments, it is necessary to balance the treated and control groups in order to estimate an unbiased causal effect of interest; otherwise, an opposite effect may be estimated, and incorrect recommendations may be given. To achieve this balance, there exist a wide variety of methods. In particular, several methods based on optimization models have been recently proposed in the causal inference literature. While these optimization-based methods showed empirical improvement over a limited number of other causal inference methods, they have not been thoroughly compared to each other and to the best causal inference methods in their relative ability to balance the pre-treatment measured covariates and to estimate the causal effect. In addition, we believe that there exist several unaddressed opportunities that operational researchers could contribute with their advanced knowledge of optimization, for the benefits of the applied researchers that use causal inference tools. In this review paper, we present an overview of the causal inference literature and describe in more detail the optimization-based causal inference methods, provide a comparative analysis of the prevailing optimization-based methods, and discuss opportunities for new methods.