Abstract: We will discuss periodic Schr\"odinger operators on the two-dimensional integer lattice. For periodic operators with small potentials, we show that the spectrum consists of at most two intervals. Moreover, there is a simple and sharp arithmetic criterion on the lattice of periods that ensures  the spectrum is an interval. Since the regime of small coupling for discrete operators mirrors the high-energy region for continuum operators, this theorem can be viewed as a discrete counterpart to the Bethe-Sommerfeld Conjecture. We will also talk about consequences for higher-dimensional operators and almost-periodic operators. [Joint work with Mark Embree]