Two recent strands of the literature on Structural Vector Autoregressions (SVARs) use higher moments for identification. One of them exploits independence and non-Gaussianity of the shocks; the other, stochastic volatility (heteroskedasticity). These approaches achieve point identification without imposing exclusion or sign restrictions. We review this work critically, and contrast its goals with the separate research program that has pushed for macroeconometrics to rely more heavily on credible economic restrictions and institutional knowledge, as is the standard in microeconometric policy evaluation. Identification based on higher moments imposes substantively stronger assumptions on the shock process than standard secondorder SVAR identification methods do. We recommend that these assumptions be tested in applied work. Even when the assumptions are not rejected, inference based on higher moments necessarily demands more from a finite sample than standard approaches do. Thus, in our view, weak identification issues should be given high priority by applied users.