Metal oxide varistor structures having a low breakdown voltage, low leakage current, high values of alpha, operational stability, and methods of making the same are disclosed. In accordance with one embodiment of the invention relating to metal oxide varistor structures, at least one of the planar surfaces of a varistor disk, for example, is provided with a recessed region for increasing the electric field intensity in the region of the recess and hence reducing the breakdown voltage of the varistor disk without altering the structural integrity of the disk. Methods for making varistor structures with one or more recesses on one or more surfaces of the varistor structures are also disclosed.