The lecture begins with a brief account of the way Maxwell arrived at his celebrated equations, and describes an alternative approach in which the concept of displacement current, often a source of difficulty to the student, is not invoked.The main part of the lecture is devoted to a discussion of the physical interpretation of some of the results which can be deduced from these equations. The close connection between electromagnetic and mechanical phenomena is discussed; for example, the theory of the deformation of an electromagnetic cavity resonator is closely linked to the conservation of angular momentum in mechanics. A surprising phenomenon exhibited by waves on a water surface is found to have an equally surprising counterpart in electromagnetic surface waves. Although the most exciting results arise when Maxwell's equations are applied to the radio-frequency part of the spectrum, surprises can occur at very low frequencies, and an example is given.In the final part of the lecture, some simple quantum ideas are introduced, and the relationship between the quantum approach and the classical approach from Maxwell's equations is briefly discussed.