We describe research and results centering on the construction and use of Bayesian models that can predict the run time of problem solvers. Our e#orts are motivated by observations of high variance in the time required to solve instances for several challenging problems. The methods have application to the decision-theoretic control of hard search and reasoning algorithms. We illustrate the approach with a focus on the task of predicting run time for general and domain-specific solvers on a hard class of structured constraint satisfaction problems. We review the use of learned models to predict the ultimate length of a trial, based on observing the behavior of the search algorithm during an early phase of a problem session. Finally, we discuss how we can employ the models to inform dynamic run-time decisions. 1