We give a characterization of Possonian domains in Rⁿ, i.e., those domains for which every bounded harmonic function is the harmonic extension of some function in L^∞ of harmonic measure. We deduce several properties of such domains, including some results of Mountford and Port. In two dimensions we give an additional characterization in terms of the logarithmic capacity of the boundary. We also give a necessary and sufficient condition for the harmonic measures on two disjoint planar domains to be mutually singular.