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.