Ever since the famous thesis of Frostman, capacities have been important in many areas of function theory. In this talk I shall be concerned only with one–variable function theory on arbitrary open subsets of the complex plane, C. It is important to stress that the open sets need not be connected. I will discuss the use of analytic capacities in connection with problems of removable singularities, holomorphic approximation, and boundary smoothness. A brief reference to the applications is in order. The connections between analytic (and harmonic) functions and the physics of perfect fluids, electrostatics, magnetostatics, classical gravitation and heat are well–known. Much of what I shall say about analytic functions (solutions of the d–bar equation) applies also to solutions of other elliptic equations, and so there are other applications, for instance to elasticity (connected to the bi-Laplacian and the d-bar-squared operator). It should also be noted that students of capacitary problems were among the first to discover and examine many of the weird–looking sets formerly regarded as pathological by most people, but now known as fractals and accepted as natural objects of study for many applications.