Although no universally accepted definition of causality exists, in practice one is often faced with the question of statistically assessing causal relationships in different settings. We present a uniform general approach to causality problems derived from the axiomatic foundations of the Bayesian statistical framework. In this approach, causality statements are viewed as hypotheses, or models, about the world and the fundamental object to be computed is the posterior distribution of the causal hypotheses, given the data and the background knowledge. Computation of the posterior, illustrated here in simple examples, may involve complex probabilistic modeling but this is no different than in any other Bayesian modeling situation. The main advantage of the approach is its connection to the axiomatic foundations of the Bayesian framework, and the general uniformity with which it can be applied to a variety of causality settings, ranging from specific to general cases, or from causes of effects to effects of causes.