This talk will start by presenting Shimizu et al's (2006) ICA-based approach (LiNGAM) for discovering acyclic (DAG) linear Structural Equation Models (SEMs) from causally sufficient, continuous-valued observational data. This is remarkable because it determines the direction of every causal arrow when no experimental data is available. Our work generalizes the above. By relaxing the acyclicity constraint, our approach, LiNG-DG, enables the discovery of arbitrary directed graph (DG) linear SEMs. We present various algorithm sketches for causal discovery with LiNG-DG, and show results of simulation for one such algorithm. When the error terms are non-Gaussian, LiNG-DG discovery algorithms output a smaller set of candidate SEMs than Richardson's Cyclic Causal Discovery (CCD) algorithm. We prove that all the models output by LiNG-DG entail the same observational distribution and are equally simple (i.e. same number of edges). This implies that without further assumptions, no algorithm can reliably narrow the set of candidate SEMs output by LiNG-DG using just observational data. However, we show that under the additional assumption of "stability", the set of candidate models output by LiNG-DG can be further narrowed down (under some conditions, to a single model).