Constrained sampling and counting are two fundamental problems in data analysis. In constrained sampling the task is to sample randomly, subject to a given weighting function, from the set of solutions to a set of given constraints. This problem has numerous applications, including probabilistic reasoning, machine learning, statistical physics, and constrained-random verification. A related problem is that of constrained counting, where the task is to count the total weight, subject to a given weighting function, of the set of solutions of the given constraints. This problem has applications in machine learning, probabilistic reasoning, and planning, among other areas. Both problems can be viewed as aspects of one of the most fundamental problems in artificial intelligence, which is to understand the structure of the solution space of a given set of constraints. This work focuses on the development of new algorithmic techniques for constrained sampling and counting, based on a universal hashing -- a classical algorithmic technique in theoretical computer science. Many of the ideas underlying the proposed approach were go back to the 1980s, but they have never been reduced to practice. Recent progress in Boolean reasoning is enabling us to reduce these algorithmic ideas to practice, and obtain breakthrough results in constrained sampling and counting, providing a new algorithmic toolbox in design verification, machine learning, probabilistic reasoning, and the like. Joint work with Supratik Chakraborty and Kuldeep S. Meel.