Social science theories often postulate systems of causal relationships among variables, which are commonly represented using directed acyclic graphs (DAGs). As non-parametric causal models, DAGs require no assumptions about the functional form of the hypothesized relationships. Nevertheless, to simplify empirical evaluation, researchers typically invoke such assumptions anyway, even though they are often arbitrary and do not reflect any theoretical content or prior knowledge. Moreover, functional form assumptions can engender bias, whenever they fail to accurately capture the true complexity of the system. In this article, we introduce causal-graphical normalizing flows (cGNFs), a novel approach to causal inference that leverages deep neural networks to empirically evaluate theories represented as DAGs. Unlike conventional methods, cGNFs model the full joint distribution of the data using a DAG specified by the analyst, without relying on stringent assumptions about functional form. This enables flexible, non-parametric estimation of any causal estimand identified from the DAG, including total effects, direct and indirect effects, and path-specific effects. We illustrate the method with a reanalysis of Blau and Duncan's (1967) model of status attainment and Zhou's (2019) model of controlled mobility. The article concludes with a discussion of current limitations and directions for future development.