Obliquely reflected diffusions in smooth domains are classical objects that have been well understood for half a century. On the other hand, many fundamental questions remain in the study of reflected diffusions in non-smooth domains, which arise in a variety of fields ranging from mathematical physics to stochastic networks. We first describe some recent results on reflected diffusions in piecewise smooth domains. We then introduce a new approach to the construction and characterization of obliquely reflected Brown motions in bounded, simply connected planar domains. The class of processes we construct also includes certain processes with jumps that have arisen in the study of SLE. This talk is partially based on joint works with Chris Burdzy, Zhenqing Chen, Weining Kang and Donald Marshall.