A major difference between elastic manifolds and elastic media is that in the latter, we must account for the possibility of topological defects. In the manifolds, the elastic fields were simply generalized ``heights'' of an interface or line, and as such had well-defined meanings as real-space coordinates. In an elastic medium, the displacements are only defined relative to some background configuration, or in the continuum limit in terms of their effect upon the density. We should therefore allow for the possibility of defects, e.g. vacancies, interstitials, and dislocations in the lattice. These have the effect of making multivalued. Configurations of the medium which include topological defects are said to involve plastic deformations, as opposed to purely elastic deformations when they are not present.