Device-to-device (D2D) communications underlaying a cellular infrastructure has been proposed as a means of taking advantage of the physical proximity of communicating devices, increasing resource utilization, and improving cellular coverage. Relative to the traditional cellular methods, there is a need to design new peer discovery methods, physical layer procedures, and radio resource management algorithms that help realize the potential advantages of D2D communications. In this article we use the 3GPP Long Term Evolution system as a baseline for D2D design, review some of the key design challenges, and propose solution approaches that allow cellular devices and D2D pairs to share spectrum resources and thereby increase the spectrum and energy efficiency of traditional cellular networks. Simulation results illustrate the viability of the proposed design.