  Fractional topological insulators are electronic systems that carry
fractionally charged excitations, conserve charge and are symmetric to reversal
of time. In this review we introduce the basic essential concepts of the field,
and then survey theoretical understanding of fractional topological insulators
in two and three dimensions. In between, we discuss the case of "two and a half
dimensions", the fractional topological insulators that may form on the two
dimensional surface of an unfractionalized three dimensional topological
insulator. We focus on electronic systems and emphasize properties of edges and
surfaces, most notably the stability of gapless edge modes to perturbations.
