The solution to many problems in applied analysis and statistics is a probability distribution (or convex function):e.g., distribution of mass, optimal strategy in a search, or worst-case distribution of errors for a particular numerical algorithm. In problems where exact analytical solutions are difficult, it is useful to have methods for generating distributions at random (which can then be used to guess at the exact solution, or analyze average-case performance). This survey lecture will describe some of the classical and recent methods for constructing a probability measure at random, including the Dubins-Freedman, sequential barycenter, and moment-interval methods. Such constructions may also be used as non-parametric priors in statistical problems, and for numerical determination of optimal parameters or design of optimal controls. The talk is aimed for the non-specialist, and will include examples and open problems.