Fractional Chern insulators (FCIs) are topologically
ordered states of interacting fermions that share their universal properties
with fractional quantum Hall states in Landau levels. FCIs have been found
numerically in a variety of two-dimensional lattice models upon partially
filling an almost dispersionless band with nontrivial topological character
with repulsively interacting fermions. I will show how FCIs emerge in bands
with Chern number C=1 and C=2 and in Z_2 topological insulators, where the
latter are accompanied by a spontaneous breaking of time-reversal symmetry.
Further, I will discuss the relevance of the noncommutative quantum geometry of
the flat topological band to the stability of FCIs and how they can be
distinguished from phases of spontaneously broken point group symmetry, such as
charge density waves.