The competing risks model considers the setting where subjects or units are exposed to multiple risks one of which may eventually cause the occurrence of the event, such as failure or recurrence or death. There is a substantial literature on identifiability and inference in both parametric and nonparametric models for competing risks. In this dissertation, we propose a parametric model for dependent competing risks that can be motivated by a frailty approach as well as by a copula approach. We establish identifiability conditions for this proposed model. We also consider competing risks regression framework and establish identifiability and methods for statistical inference in this framework. This proposed model has been further extended to analysis of semi-competing data while we again establish identifiability and statistical inference. The proposed models have been illustrated in extensive simulation studies and we apply these models to analyze competing risks data from a Tamoxifen trial on breast cancer patients and to analyze semi-competing risks data from a trial on tuberculous pericarditis collected in eight countries in Africa.