Interactions and phase transition in Dirac systems in two and three dimensions

I will review recent advances in our understanding of the effects of electron-electron interactions in systems that exhibit band crossing or touching at the Fermi level. Two study cases will be covered in detail: 1) interacting electrons on the honeycomb or pi-flux lattice,  2) and the electronic systems with the quadratic band touching in two and three dimensions, such as bilayer graphene and mercury telluride, for example. I will discuss the universal physics of the semimetal - Mott insulator transition in the Hubbard model for the former case, as uncovered by field-theoretic and Monte Carlo methods. Three dimensional systems with the quadratic band touching will be argued to be a precious examples of the possible non-Fermi liquid physics, ultimately giving way to new broken symmetry states at low temperatures.