"We propose to study topologically ordered phases of matter, especially those that are of potential use for topological quantum computation.The research proposed is organized along two intertwined lines.1. The first line focuses on the physical realization of topologically ordered phases in systems of bosons or fermions on a two dimensional lattice, subject to strong real or ""synthetic"" magnetic fields.We plan to study these systems using the analytical and numerical tools which have been developed for the study of the two dimensional electron liquids of the fractional quantum Hall effect (FQHE). The effects of the lattice on the known FQH states will be studied, and newly emerging states which can only exist in the presence of a lattice, will be explored. These states may in principle be realized in ultrcold Bose and Fermi gases subject to an optical lattice potential.2. The second line of research deals with the detection and characterization of topological order, initially in models, but with a view to experiment, using tools based on quantum entanglement of many body states, such as the topological entanglement entropy.Concrete plans include numerical calculation of the entanglement entropy in a variety of systems, using a variational Monte Carlo code based on the so called replica method. This method has not been tried before in this context and promises to allow for larger system sizes and better extrapolation to the thermodynamic limit (this has been a problem with earlier calculations).After extensive testing on simpler models, we plan to apply these new tools to the study of fractional quantum Hall systems, especially the lattice FQH systems studied in research line 1."