We propose a novel class of dynamic shrinkage processes for Bayesian time series and regression analysis. Building upon the global-local framework of prior construction, in which continuous scale mixtures of Gaussian distributions are employed for both desirable shrinkage properties and computational tractability, we allow the local scale parameters to depend on the history of the shrinkage process. The resulting processes inherit the desirable shrinkage behavior of popular global-local priors, such as the horseshoe, but provide additional localized adaptivity, which is important for modeling time series data or regression functions with local features. We construct an efficient Gibbs sampling algorithm by adapting successful techniques from stochastic volatility modeling and deriving a Polya-Gamma scale mixture representation of the proposed process. We use dynamic shrinkage processes to develop an adaptive time-varying parameter regression model for the Fama-French five-factor asset pricing model. Our dynamic analysis of manufacturing and healthcare industry data shows that with the exception of the market risk, no other risk factors are significant except for brief periods.