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 processes inherits the desirable shrinkage behavior of popular global-local priors, such as the horseshoe prior, but provides additional localized adaptivity, which is important for modeling time series data or regression functions with local features. Efficient Gibbs sampling algorithms are employed via techniques from stochastic volatility modeling and by deriving a Polya-Gamma scale mixture representation of the process. Dynamic shrinkage processes produce superior Bayesian trend filtering estimates and posterior intervals for irregular curve-fitting of minute-by-minute Twitter CPU usage data. We develop an adaptive time-varying parameter regression model to assess the efficacy of the Fama-French 5-factor asset pricing model with momentum added as a 6th factor.