Flexible parametric distribution models that can represent both skewed and symmetric distributions, namely skew symmetric distributions, can be constructed by skewing symmetric kernel densities by using weighting distributions. In this book, we study a multivariate skew family that have either centrally symmetric or spherically symmetric kernel. Specifically, we define multivariate skew symmetric forms of uniform, normal, Laplace, and logistic distributions by using the cumulative distribution functions of the same distributions as weighting distributions. Matrix and array variate extensions of these distributions are also introduced herein. We propose an estimation procedure based on the maximum product of spacings method and model identification. This idea also leads to bounded model selection criteria that can be considered as alternatives to Akaike's and other likelihood based criteria when the unbounded likelihood may be a problem. Applications of skew symmetric distributions to data are also considered.