Archaeologists are increasingly becoming aware of an approach to data investigation known as Bayesian statistics. In this paper we outline both the philosophical and statistical background to the approach. We show that it provides a logical and coherent framework in which to make inferences on the basis of both data and a priori expert knowledge. We note that adoption of the Bayesian framework is particularly timely since there have been recent dramatic developments in numerical methods which mean that a number of previous implementation problems have now been solved. As a result, many questions of archaeological interest, which require the use of complex statistical models, are being investigated using this methodology. We use a variety of recently published examples from a range of archaeological areas to illustrate the type of questions that can be answered and the nature of the methodologies used, and we make comparisons with the results obtained using more traditional statistical techniques.