Existing methods that assume non-Gaussian distributions to identify parameters and conduct inference in SVAR models work poorly when the deviations from Gaussianity are small. In particular, confidence bands for the impulse responses suffer from coverage distortions. We propose a robust and efficient semi-parametric approach to conduct hypothesis tests and compute confidence bands in the SVAR model. The method exploits non-Gaussianity when it is present, but yields correct coverage regardless of the distance to the Gaussian distribution. We evaluate the method in a simulation study and revisit several empirical studies to highlight the practical relevance of our methodology and the limitations of assuming non-Gaussianity for identification.