he selection of an appropriate model from a potentially large class of candidate models is an issue that is central to regression. The object of this thesis is to connect many different aspects of the growing model selection field by examining the different lines of reasoning that have motivated the derivation of both classical and modern criteria, and then to examine the performance of these criteria. In this way, we hope to bridge theory and practicality. We begin to understand the different approaches that inspired the many criteria considered in this thesis by presenting some of the most commonly used selection criteria. Then, we give a corrected version of the KIC criterion for model selection in settings where the sample size is small or when it is large and the dimension of the candidate model is relatively small. In such case, the KIC criterion provides a large negative bias. We will present some corrected versions criteria for multiple and multivariate regression models to overcome this problem. Its signal-tonoise ratio will be pointed out in each case. The performance of new criteria relative to other criteria bias, these estimators can achieve a greater reduction in variance and an overall reduction in MSE....