Smoothly time-varying (TV) GARCH models via an asymmetric logistic function mechanism are proposed, which are incorporated into the conditional volatility equation for capturing smooth structural breaks in the volatility of financial time series. The proposed models allow smooth transitions of varying “speed” between multiple, persistent regimes. A Bayesian computational method is employed to identify the locations of smooth structural transitions, and for estimation and inference, simultaneously accounting for heteroskedasticity. An informative prior is proposed to help ensure identification and allow accurate inference. The proposed methods are illustrated using simulated data, and an empirical study provides evidence for significant improvements in fit for the proposed smooth asymmetric time-varying volatility TV-GARCH models in two international stock market return series. A forecast study shows the proposed models significantly add to forecast accuracy for both volatility and Value-at-Risk.