One of the most important concepts of relativistic physics is that of a world line, i.e., the space-time trajectory of a point particle. In Lorentzian geometry, it is modelled by the so-called causal curve, and the questions concerning which spacetime points can be connected by means of causal curves lead to a vast area of study known as causality theory. In the talk, I will present how the basic notions of causality theory can be naturally extended to probability measures on spacetimes (what is motivated by both classical and quantum physics). In particular, I will discuss the notion of a causal evolution of probability measures, its deep connection with the continuity equation (known from elementary physics) and a surprisingly nice topological properties of the space of causal curves.