  We present a pedagogical review of the physics of fractional Chern insulators
with a particular focus on the connection to the fractional quantum Hall
effect. While the latter conventionally arises in semiconductor
heterostructures at low temperatures and in high magnetic fields, interacting
Chern insulators at fractional band filling may host phases with the same
topological properties, but stabilized at the lattice scale, potentially
leading to high-temperature topological order. We discuss the construction of
topological flat band models, provide a survey of numerical results, and
establish the connection between the Chern band and the continuum Landau
problem. We then briefly summarize various aspects of Chern band physics that
have no natural continuum analogs, before turning to a discussion of possible
experimental realizations. We close with a survey of future directions and open
problems, as well as a discussion of extensions of these ideas to higher
dimensions and to other topological phases.
