Magnetic phenomena are commonly used for data-storage applications, but there are relatively few attempts to exploit magnetic phenomena for logic applications. In this presentation, we review our recent work on logic functionality in systems of physically-coupled nanometer-scale magnets. Nanomagnets are special in that they exhibit only two distinct stable states of magnetization, which is the basis for their use in magnetic random access memories and in hard disk drives. These latter applications, for which fabrication of dense arrays is under development, face the challenge of avoiding magnetic dipole interactions between individual elements, which limits data storage density. In contrast, these interactions are exploited in the magnetic quantum-dot cellular automata (MQCA) system, which is a network of closely-spaced, dipole-coupled, single-domain nanomagnets designed for digital computation. Our work demonstrates that logic functions can be realized in properly-structured arrays of physically-coupled nanomagnets, and points to the possibility of all-magnetic information processing systems that incorporate both memory and logic.