This book presents a simple graphical theory unifying causal directed acyclic graphs (DAGs) and potential (aka counterfactual) outcomes via a node-splitting transformation. It introduces a new graph, the Single-World Intervention Graph (SWIG). The SWIG encodes the counterfactual independences associated with a specific hypothetical intervention on the set of treatment variables. The nodes on the SWIG are the corresponding counterfactual random variables. The theory is illustrated with a number of examples. The authors' graphical theory of SWIGs may be used to infer the counterfactual independence relations implied by the counterfactual models developed in Robins (1986, 1987). Moreover, in the absence of hidden variables, the joint distribution of the counterfactuals is identified; the identifying formula is the extended g-computation formula introduced in (Robins et al., 2004). Although Robins (1986, 1987) did not use DAGs we translate his algebraic results to facilitate understanding of this prior work. An attractive feature of Robins' approach that it largely avoids making counterfactual independence assumptions that are experimentally untestable.As an important illustration, this book revisits the critique of Robins' g-computation given in (Pearl, 2009, Ch. 11.3.7); it uses SWIGs to show that all of Pearl's claims are either erroneous or based on misconceptions. It also shows that simple extensions of the formalism may be used to accommodate dynamic regimes, and to formulate non-parametric structural equation models in which assumptions relating to the absence of direct effects are formulated at the population level. Finally, this book shows that the authors' graphical theory also naturally arises in the context of an expanded causal Bayesian network in which we are able to observe the natural state of a variable prior to intervention.