In this talk, I would review some recent developments in FQH states and Partial filled Landau Levels. Starting with the geometric theory in FQH states, I would first introduce the nematic FQH theory where rotation symmetry breaking phase coexists with the Chiral topological order. The nematic order parameter displays exotic interplay with topological quasiparticle and EM response, as a consequence of the Wen-Zee effect and gravitational anomaly.
In the second part of my talk, I would move on to compressibly FQH states in the 0th LL, e.g. the half-filled Landau Levels as a composite Fermi liquid. I would briefly review the Dirac theory description for half-filled Landau Levels raised by Son, and propose some measurements to verified the hidden Berry phase of the Dirac fermi surface in such system.