We address the problem of dynamic variable selection in time series regression, where the set of active predictors is allowed to evolve over time. To capture time-varying variable selection uncertainty, we introduce new dynamic shrinkage priors for the time series of regression coefficients. These priors are characterized by two main ingredients: smooth parameter evolutions and intermittent zeroes for modeling predictive breaks. More formally, our proposed Dynamic Spike-and-Slab (DSS) priors are constructed as mixtures of two processes: a spike process for the irrelevant coefficients and a slab autoregressive process for the active coefficients. The mixing weights are themselves time-varying and depend on a lagged value of the series. We characterize dynamic selection thresholds for MAP smoothing and implement a one-step-late EM algorithm for fast calculations. We demonstrate, through simulation and a topical high-dimensional macroeconomic dataset, that DSS priors are far more effective at finding signal and improving forecasts compared to other existing strategies.