  Fractional Chern insulators (FCIs) are lattice generalizations of the
conventional fractional quantum Hall effect (FQHE) in two-dimensional (2D)
electron gases. They typically arise in a 2D lattice without time-reversal
symmetry when a nearly flat Bloch band with nonzero Chern number is partially
occupied by strongly interacting particles. Band topology and interactions
endow FCIs exotic topological orders which are characterized by the precisely
quantized Hall conductance, robust ground-state degeneracy on high-genus
manifolds, and fractionalized quasiparticles. Since in principle FCIs can exist
at zero magnetic field and be protected by a large energy gap, they provide a
potentially experimentally more accessible avenue for observing and harnessing
FQHE phenomena. Moreover, the interplay between FCIs and lattice-specific
effects that do not exist in the conventional continuum FQHE poses new
theoretical challenges. In this chapter, we provide a general introduction of
the theoretical model and numerical simulation of FCIs, then pay special
attention on the recent development of this field in moir\'e materials while
also commenting on potential alternative implementations in cold atom systems.
With a plethora of exciting theoretical and experimental progress, topological
flat bands in moir\'e materials such as magic-angle twisted bilayer graphene on
hexagonal boron nitride have indeed turned out to be a remarkably versatile
platform for FCIs featuring an intriguing interplay between topology, geometry,
and interactions.
