This chapter gives a careful formal development of Chain Event Graph (CEG) models. It illustrates how useful the probability tree and the CEG can be in the depiction of a given problem. The chapter prepares the framework for showing that these graphs are equally useful from a probabilistic perspective. It shows how to draw out an explanation of a probability tree or CEG in natural language. The chapter provides a formal analysis of statistical models represented by CEGs. It determines which types of events can be depicted unambiguously within a CEG. The chapter relates the colouring and the graph of a CEG to distributional assumptions on the underlying statistical model and, in particular, to their implied conditional independence assumptions. It shows how to draw out an explanation of a probability tree or CEG in natural language. The chapter utilizes incident- and vertex-random variables in a CEG to read various implicit conditional independence assumptions from the graph.