Equations of motion and the Lagrangian formalism for extended objects coupled to Abelian and non-Abelian gauge fields are developed. These equations are minimal generalizations of the corresponding equations for point particles. It is seen that the string superconducts when it couples to an Abelian gauge field. Further, in this case, (a) the total charge on it is quantized, and (b) the total magnetic flux through it is quantized and conserved if it is closed and no segment of it is electrically neutral. The Lagrangians which lead to the equations of motion are not unique. Here, for a suitable Lagrangian, property (a) emerges at the classical level, the total charge being a topological invariant which labels the elements of 1[U(1)]. Both of these properties partially generalize to other extended objects and non-Abelian gauge fields. It is pointed out that for some Lagrangians, extended objects may have topological invariants (the analogs of total charge) for any gauge group. Supersymmetric extensions of the interaction Lagrangians are also outlined. For a point particle, such an extension correctly describes a spin-half particle in an Abelian or a non-Abelian gauge field. © 1979 The American Physical Society.