The name "Chern insulator" refers to an insulator that exhibits integer and even fractional quantum Hall effect with no net magnetic field. We show that the topologically nontrivial bands of Chern insulators are adiabatic cousins of the Landau bands of Hofstadter lattices. We demonstrate adiabatic connection also between the familiar fractional quantum Hall states on Hofstadter lattices and the fractional Chern insulator states in partially filled Chern bands, which implies that they are in fact different manifestations of the same phase. This adiabatic path provides a way of generating many more fractional Chern insulator states, and helps clarify that non-uniformity in the distribution of the Berry curvature is responsible for weakening or altogether destroying fractional topological states. (Work in collaboration with Y. H. Wu and K. Sun.)