In the 1920's the geneticist Sewall Wright introduced a class of Gaussian statistical models represented by graphs containing directed and bi-directed edges, known as path diagrams. These models have been used extensively in psychometrics and econometrics where they are called structural equation models. I will first describe the subclass of bow-free acyclic path diagrams, which have desirable statistical properties. I will then characterize a subclass of models that are characterized by their Markov properties. Lastly I will outline recent work aimed at characterizing non-Markovian constraints that may arise. (This is joint work with Mathias Drton, Michael Eichler and Masashi Miyamura.)