In 2010 it was discovered that placing graphene sheets on hexagonal boron nitride (hBN) substrate improved their quality significantly, paving the way for many new experiments on high-mobility graphene. Until recently, however, little attention was paid to the effect of the hBN substrate on the electronic structure of graphene. In this talk, I will discuss recent experiments at MIT in which the hBN substrate has been found to induce a sizable band gap at the charge-neutrality point in monolayer graphene, imparting a mass to the normally massless Dirac charge carriers. The bandgap occurs only in samples in which the twist angle between the graphene and hBN sheets is small, producing a long-wavelength moiré that acts as a superlattice potential; by adjusting the twist angle the bandgap can be tuned. The moiré superlattice potential also allows us to study the problem of a charged particle in a periodic potential and magnetic field – the so-called Hofstadter problem – whose theoretical solution exhibits a rare instance of fractal behavior in a quantum-mechanical energy spectrum.I will also discuss our recent studies of the phase diagram of charge-neutral monolayer graphene, where we find we can induce a transition from the canted antiferromagnetic phase to one displaying a quantum spin Hall effect.