We show that structural smooth transition vector autoregressive models are statistically identified if the shocks are mutually independent and at most one of them is Gaussian. This extends a known identification result for linear structural vector autoregressions to a time-varying impact matrix. We also propose an estimation method, show how a blended identification strategy can be adopted to address weak identification, and establish a sufficient condition for ergodic stationarity. The introduced methods are implemented in the accompanying R package sstvars. Our empirical application finds that a positive climate policy uncertainty shock reduces production and raises inflation under both low and high economic policy uncertainty, but its effects, particularly on inflation, are stronger during the latter.