We define a general family of multivariate skew-symmetric distributions which includes generalized skew-elliptical distributions as a special case. In particular, it also includes the multivariate skew-normal, skew-t, skew-Cauchy and skew-elliptical distributions. We show that any multivariate pdf admits a skew-symmetric representation and study several characteristics of this representation. We establish various invariance properties for quadratic forms in skew-symmetric random vectors with links to chi-square distributions. These properties imply that standard inferential methods might be misleading when applied to time series and spatial processes with skew-symmetric distributions. However, the same property is beneficial for inference from non-random (biased) samples. We also propose a flexible class of skew-symmetric distributions by constructing an enumerable dense subset of skewing functions. This flexible family of distributions can capture skewness, heavy tails, and multimodality systematically. Moreover, it is straightforward to simulate pseudo-realizations from this family. This is an attractive property for applications requiring EM or MCMC implementations. We provide several examples and applications for illustration. In particular, we relax the standard normal assumption of the random effects in linear mixed models to flexible generalized skew-elliptical distributions.