A two-dimensional electron system subjected to a large perpendicular magnetic field is a veritable goldmine for the condensed matter physicist. The original discovery of the integer quantized Hall effect (IQHE) in semiconductor heterostructures was followed by the even more unexpected discovery of its fractional counterpart (FQHE) soon thereafter. In the three decades that have elapsed since those Nobel Prize winning discoveries, this system has continued to provide an increasing number of exotic phenomena that have literally revolutionized our understanding of condensed matter. Examples include fractionalization of the electronic charge, composite particles, Abelian and non-Abelian quantum states, and topological spin excitations, in addition to charge-density-wave phases, including electron crystals, bubble and striped phases. After an introduction to the rich and fascinating world of two-dimensional electrons, I will describe two recent efforts in our continual search for new phenomena. First, I will consider electrons in the two-dimensional material graphene, where the band dispersion is of the relativistic, Dirac form, instead of the parabolic non-relativistic spectrum in Gallium Arsenide. Then, I will discuss the effect of mass anisotropy, which exists in systems based on many-valley semiconductors such as AlAs, Si and Ge, or in isotropic GaAs when the magnetic field is tilted away from the plane normal; this has resulted in a new understanding of the FQHE. Using exact numerical many-body calculations for both types of systems, I will discuss how these systems add to the richness and diversity of quantum Hall phases, including (i) the possibility of optimizing properties, and (ii) uncovering universal chiral Luttinger liquid behavior, which has remained elusive for semiconductor based electron gases in the quantum Hall regime.