In a previous work the dynamics of relativistic extended objects (i.e., strings, shells, etc.) coupled to Abelian or non-Abelian gauge fields was developed. The extended objects possessed an electriclike current which was defined in the associated Lie algebra of the gauge group under consideration. In the present paper, the interaction between the extended objects and gauge fields is slightly modified so that the objects behave like superconductors. By this we mean (a) the electrical conductivity is infinite and (b) for objects other than strings, a magnetic shielding or Meissner effect (with zero penetration depth) is present. Both (a) and (b) are features which occur in the classical description of the system. We also develop the dynamics for a system which is dual to the one described above. That is, instead of possessing an electric current, the objects here carry a magnetic current (Abelian or non-Abelian). Furthermore, the magnetic conductivity is infinite, and for objects other than strings an electric shielding or "dual" Meissner effect is present. The systems developed here contain Dirac's extended electron model and the MIT bag model as special cases. The former coincides with the description of an electrically charged shell. In the latter, we verify that the dynamics of a cavity within a (magnetic) superconducting vacuum is identical to that of a glueball in the MIT bag. This agrees with the view that the true quantum-chromodynamic (QCD) vacuum may be in a magnetic superconducting phase, and that the "dual" Meissner effect may be relevant for the confinement question. We also examine the possibility of the QCD vacuum being in an electric (or conventional) superconducting phase and a mixed superconducting phase, and comment on the confinement question for these two cases. © 1982 The American Physical Society.