We develop a new algorithm for inference in structural vector autoregressions (SVARs) identified with sign restrictions that can accommodate big data and modern identification schemes. The key innovation of our approach is to move beyond the traditional accept-reject framework commonly used in sign-identified SVARs. We show that an elliptical slice within Gibbs sampler can deliver dramatic gains in computational speed and render previously infeasible applications tractable. We also prove that the algorithm is well-defined, in the sense that its stationary distribution coincides with the posterior distribution of interest. To illustrate the approach in the context of sign-identified SVARs, we use a tractable example. We further assess the performance of our algorithm through two applications: a well-known small-SVAR model of the oil market featuring a tight identified set, and a large SVAR model with more than ten shocks and 100 sign restrictions.