In this chapter, we extend the standard Gaussian stochastic volatility Bayesian VAR by employing the generalized hyperbolic skew Student’s t distribution for the innovations. Allowing the skewness parameter to vary over time, our specification permits flexible modelling of innovations in terms of both fat tails and—potentially dynamic—asymmetry. In an empirical application using US data on industrial production, consumer prices and economic policy uncertainty, we find support—although to a moderate extent—for time-varying skewness. In addition, we find that shocks to economic policy uncertainty have a negative effect on both industrial production growth and CPI inflation.