This dissertation develops a novel stochastic tree ensemble method for nonlinear regression, which I refer to as XBART, short for Accelerated Bayesian Additive Regression Trees. By combining regularization and stochastic search strategies from Bayesian modeling with computationally efficient techniques from recursive partitioning approaches, the new method attains state-of-the-art performance: in many settings it is both faster and more accurate than the widely-used XGBoost algorithm. Via careful simulation studies, I demonstrate that our new approach provides accurate point-wise estimates of the mean function and does so faster than popular alternatives, such as BART, XGBoost and neural networks (using Keras). This dissertation also prove a number of basic theoretical results about the new algorithm, including consistency of the single tree version of the model and stationarity of the Markov chain produced by the ensemble version. Furthermore, I demonstrate that initializing standard Bayesian additive regression trees Markov chain Monte Carlo (MCMC) at XBART-fitted trees considerably improves credible interval coverage and reduces total run-time.