While the theory of causality is widely viewed as an extension of probability theory, a view which we share, there was no universally accepted, axiomatic framework for causality analogous to Kolmogorov's measure-theoretic axiomatization for the theory of probabilities. Instead, many competing frameworks exist, such as the structural causal models or the potential outcomes framework, that mostly have the flavor of statistical models. To fill this gap, we propose the notion of causal spaces, consisting of a probability space along with a collection of transition probability kernels, called causal kernels, which satisfy two simple axioms and which encode causal information that probability spaces cannot encode. The proposed framework is not only rigorously grounded in measure theory, but it also sheds light on long-standing limitations of existing frameworks, including, for example, cycles, latent variables, and stochastic processes. Our hope is that causal spaces will play the same role for the theory of causality that probability spaces play for the theory of probabilities.