Four new generalizations of power-Cauchy distribution are proposed in this thesis. Firstly, the Poisson-X family is proposed and its important mathematical properties are obtained. Then, a member of the Poisson-X family namely the Poisson power-Cauchy)distributionisdefinedanditsstructuralpropertiesareinvestigated. The proposed model is fitted to three real-life data sets to illustrate its flexibility. Secondly, the log-odd normal generalized family of distributions is introduced. Then, a special model of this family, the log-odd normal power-Cauchy is defined and its mathematical properties are obtained. A real-life data set is used to prove the superiority of the proposed model. Thirdly, the Weibull-Power-Cauchy distribution is proposed and its mathematical properties are obtained. Two useful characterizations based on truncated moments are also presented. The proposed model is applied to three real-life data sets to investigate its flexibility. Lastly, a new extensionofpower-Cauchymodelisproposedbycompoundingthepower-Cauchyand negative-binomial distribution called the power-Cauchy negative-binomial distribution. Some mathematical properties of the proposed model are obtained and the model parameters are estimated using the maximum likelihood method. A simulation study is carried out to investigate the performance of maximum likelihood method. The flexibility of the proposed model is illustrated through three real-life data sets. Then, a new class of regression model is introduced for location and scale based on the logarithm of the proposed random variable and, estimation and inference on the regression coefficients are discussed.