Two approaches to characterizing reflected diffusions include the submartingale problem formulation or a formulation in terms of the Skorokhod problem and stochastic differential equations. We introduce these formulations for a large class of piecewise smooth domains and, under suitable assumptions, we show that well-posedness of the submartingale problem is equivalent to existence and uniqueness in law of weak solutions to the corresponding stochastic differential equation with reflection. Our result generalizes to the case of reflecting diffusions a classical equivalence result due to Stroock and Varadhan between stochastic differential equations and martingale problems. The analysis in the case of reflected diffusions in domains with non-smooth boundaries is considerably more subtle and requires a careful analysis of the behavior of the reflected diffusion on the boundary of the domain. In particular, we show that the equivalence can fail to hold when our assumptions are not satisfied This is joint worth with Weining Kang.