In this talk, we consider causal discovery when the underlying structure corresponds to a linear structural equation model with error terms which are non-Gaussian. Previous work by Shimizu et al. (2006) has shown that under this framework, a unique directed acyclic graph--not simply an equivalence class--can be identified from infinite data. We extend that result in two directions. First, we show that a unique graph can still be consistently recovered in the high dimensional setting where p, the number of variables, exceeds n, the number of observed samples. In particular, we present a tractable algorithm which will consistently recover the true graph when it is suitably sparse and the sample moments of the data concentrate exponentially. Finally, time permitting, we will show that the identifiability results also extend to models which allow for unobserved confounding, albeit in a structured way. Specifically, we show when the data is generated by a model corresponding to a bow-free acyclic path (BAP) diagram, the exact BAP--not simply an equivalence class--can be recovered consistently.