  I review why and how physical states with fractional quantum numbers can
occur, emphasizing basic mechanisms in simple contexts. The general mechanism
of charge fractionalization is the passage from states created by local action
of fields to states having a topological character, which permits mixing
between local and topological charges. The primeval case of charge
fractionalization for domain walls, in polyacetylene and more generally, can be
demonstrated convincingly using Schrieffer's intuitive counting argument, and
derived formally from analysis of zero modes and vacuum polarization. An
important generalization realizes chiral fermions on high-dimensional domain
walls, in particular for liquid He3 in the A phase. In two spatial dimensions,
fractionalization of angular momentum and quantum statistics occurs, for
reasons that can be elucidated both abstractly, and specifically in the context
of the quantum Hall effect.
