A comprehensive methodology for inference in vector autoregressions (VARs) using sign and other structural restrictions is developed. The reduced-form VAR disturbances are driven by a few common factors and structural identification restrictions can be incorporated in their loadings in the form of parametric restrictions. A Gibbs sampler is derived that allows for reduced-form parameters and structural restrictions to be sampled efficiently in one step. A key benefit of the proposed approach is that it allows for treating parameter estimation and structural inference as a joint problem. An additional benefit is that the methodology can scale to large VARs with multiple shocks, and it can be extended to accommodate non-linearities, asymmetries, and numerous other interesting empirical features. The excellent properties of the new algorithm for inference are explored using synthetic data experiments, and by revisiting the role of financial factors in economic fluctuations using identification based on sign restrictions.