  Topological insulators and their intriguing edge states can be understood in
a single-particle picture and can as such be exhaustively classified.
Interactions significantly complicate this picture and can lead to entirely new
insulating phases, with an altogether much richer and less explored
phenomenology. Most saliently, lattice generalizations of fractional quantum
Hall states, dubbed fractional Chern insulators, have recently been predicted
to be stabilized by interactions within nearly dispersionless bands with
non-zero Chern number, $C$. Contrary to their continuum analogues, these states
do not require an external magnetic field and may potentially persist even at
room temperature, which make these systems very attractive for possible
applications such as topological quantum computation. This review recapitulates
the basics of tight-binding models hosting nearly flat bands with non-trivial
topology, $C\neq 0$, and summarizes the present understanding of interactions
and strongly correlated phases within these bands. Emphasis is made on
microscopic models, highlighting the analogy with continuum Landau level
physics, as well as qualitatively new, lattice specific, aspects including
Berry curvature fluctuations, competing instabilities as well as novel
collective states of matter emerging in bands with $|C|>1$. Possible
experimental realizations, including oxide interfaces and cold atom
implementations as well as generalizations to flat bands characterized by other
topological invariants are also discussed.
