  The study of topology in solids is undergoing a renaissance following renewed
interest in the properties of ferroic domain walls as well as recent
discoveries regarding topological insulators and skyrmionic lattices. Each of
these systems possess a property that is `protected' in a symmetry sense, and
is defined rigorously using a branch of mathematics known as topology. In this
article we review the formal definition of topological defects as they are
classified in terms of homotopy theory, and discuss the precise
symmetry-breaking conditions that lead to their formation. We distinguish
topological defects from geometric defects, which arise from the details of the
stacking or structure of the material but are not protected by symmetry. We
provide simple material examples of both topological and geometric defect
types, and discuss the implications of the classification on the resulting
material properties.
