The entanglement entropy associated to a region of space plays an important role in black hole thermodynamics, quantum information theory, condensed matter theory and the AdS/CFT correspondence. In a gauge theory the entanglement entropy acquires qualitatively new features, and its definition requires introducing edge states of the kind that appear in three-dimensional quantum gravity and the quantum Hall effect. The edge states transform nontrivially under the gauge group of the boundary, which constrains the form of the reduced density matrix. I will illustrate this explicitly using a lattice regulator, and show that the gauge symmetry leads to a splitting of the entanglement entropy into bulk and boundary parts. Although the considerations are very general, I will illustrate them with three examples: Yang-Mills theory in two dimensions, the toric code, and lattice Yang-Mills at strong coupling.