We review recent developments in applying Bayesian probabilistic and statistical ideas to expert systems. Using a real, moderately complex, medical example we illustrate how qualitative and quantitative knowledge can be represented within a directed graphical model, generally known as a belief network in this context. Exact probabilistic inference on individual cases is possible using a general propagation procedure. When data on a series of cases are available, Bayesian statistical techniques can be used for updating the original subjective quantitative inputs, and we present a set of diagnostics for identifying conflicts between the data and the prior specification. A model comparison procedure is explored, and a number of links made with mainstream statistical methods. Details are given on the use of Dirichlet prior distributions for learning about parameters and the process of transforming the original graphical model to a junction tree as the basis for efficient computation.