In this thesis , we present an expanded account of skew-symmetric distributions based on articles by Volkmer and Hamedani (2007), Sartori (2006) and Azzalini (1985). There has been an increasing interest in finding more flexible methods to represent features of the data as adequately as possible and to reduce unrealistic assumptions. The motivation originate from data sets that often do not satisfy some standard assumptions such as normality. Therefore, the construction of parametric families of asymmetric distributions whichare analytically tractable, can accommodate practical values of skewness and kurtosis, andstrictly include the normal distribution, can be useful for data modeling, statistical analysis,and robustness studies of normal theory methods. Although the idea of modeling skewness by means of the construction of a mathematically tractable family including the normal distribution was proposed early by other authors (see, for example, O’Hagan and Leonard (1976)), the , formal definition of the univariate skew-symmetric family is due to Azzalini(1985). According to the definition which he proposed, a random variable Y has a skew-symmetric distribution with skewness parameter ?, if its density is density is f ? (y) =2f(y)G(?y), y,? ?R, where f is continuous distribution which is symmetric about 0 and let G be the distribution function of a distribution that is absolutely continuous and symmetric about 0.This family of distributions, most of which have non-zero skew, contains the symmetric density f, as is evidenced by the choice of ?=0. Thus, we have a model which allows for a bit of variation about the symmetric model. In this thesis, the parameter and even moments do not depend on this parameter. A characterization of a symmetric probability density function based on certain properties of its associated skew family is presented. Skew-normal model is one of such distributions that are skew and yet possess many properties of the normal distribution. In this research, the problem with the method of moment, maximum likelihood and the maximum likelihood estimation with a modify function for the one parameter skew-normal distribution are analyzed. Finally, two conventional methods of goodness of fit test for the skew-normal distribution are discussed.