Large Bayesian VARs with stochastic volatility are increasingly used in empirical macroeconomics. The key to make these highly parameterized VARs useful is the use of shrinkage priors. We develop a family of priors that captures the best features of two prominent classes of shrinkage priors: adaptive hierarchical priors and Minnesota priors. Like the adaptive hierarchical priors, these new priors ensure that only ‘small’ coefficients are strongly shrunk to zero, while ‘large’ coefficients remain intact. At the same time, these new priors can also incorporate many useful features of the Minnesota priors, such as cross-variable shrinkage and shrinking coefficients on higher lags more aggressively. We introduce a fast posterior sampler to estimate BVARs with this family of priors - for a BVAR with 25 variables and 4 lags, obtaining 10,000 posterior draws takes about 3 minutes on a standard desktop. In a forecasting exercise, we show that these new priors outperform both adaptive hierarchical priors and Minnesota priors.