We present some recent results on real-variable BMO, VMO and their nonhomogeneous ("localized") counterparts. This includes joint work with Almut Burchard (Toronto) and Ryan Gibara (Cincinnati) on the boundedness and continuity of the decreasing and symmetric decreasing rearrangement operators acting on rearrangeable functions in BMO and VMO, as well as joint work with Almaz Butaev (Cincinnati) identifying locally uniform domains as extension domains for Goldberg's bmo.