Certain fractional quantum Hall wavefunctions -- particularly including the Laughlin, Moore-Read, and Read-Rezayi wavefunctions -- have special structure that makes them amenable to analysis using an exeptionally wide range of techniques including conformal field theory (CFT), thin cylinder or torus limit, study of symmetric polynomials and Jack polynomials, and so-called ``special parent Hamiltonians. This review discusses these techniques as well as explaining to what degree some other quantum Hall wavefunctions share this special structure. Along the way we will explore the physics of quantum Hall edges, entanglement spectra, quasiparticles, nonabelian braiding statistics, and Hall viscosity, among other topics. As compared to a number of other recent reviews, most of this review is written so as to {it not} rely on results from conformal field theory -- although a short discussion of a few key relations to CFT are included near the end.
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