A core tenet of condensed matter physics has been that different phases of matter can be classified according to Landau's symmetry breaking paradigm. It has become clear, however, that phases of matter exist that are not distinguished by symmetry, but rather by topology. A paradigmatic example of this topological order are the fractional quantum Hall phases, which are the topic of this dissertation. Such phases exhibit the fractional quantum Hall effect, which occurs when electrons confined to two dimensions are subjected to a strong perpendicular magnetic field at very low temperatures. Characterized by a precise quantization of the Hall resistance and a concomitant vanishing of the longitudinal resistance, the fractional quantum Hall effect results from the formation of a strongly correlated quantum liquid of electrons. This quantum liquid supports remarkable quasiparticle excitations, which carry a fractional charge and are thought to obey fractional statistics beyond the familiar Bose-Einstein and Fermi-Dirac statistics.

The theoretical understanding of the topological orders realized by the fractional quantum Hall states has progressed by the proposal of explicit trial wave functions as well as various types of effective field theories. This dissertation focuses on two series of trial wave functions, abelian and non-abelian hierarchy wave functions. We study the non-abelian hierarchy wave functions using conformal field theory techniques, by means of which the associated topological properties are studied. These include the fractional charges of the quasiparticles and their non-abelian fractional statistics. In addition, we study abelian hierarchy wave functions using effective Ginzburg-Landau theories in a way that connects to their known conformal field theory description.