In this Thesis, I provide a theoretical description of the properties of graphene on atomically flat hexagonal substrates, such as hexagonal boron nitride (hBN). It is known that the electronic properties of graphene-based devices can be dramatically improved by the use of such substrates. At the same time, a small lattice mismatch or misalignment angle, results in the formation of the large quasi-periodic structure known as a moire pattern. The dominant effect of this, on graphene's electrons, can be described in terms of scattering by the simplest harmonics of the moire pattern, which, combined with the symmetry of the system, allows a generic phenomenological Hamiltonian to be written. We systematically investigate the characteristic features that appear in the resulting miniband spectrum, and show that there generally exists additional secondary Dirac points, isolated on the energy axis, on the edge of the first moire miniband. This analysis is extended to bilayer-graphene/hBN heterostructures, which generically feature a gap at the edge of the first moire miniband. In a strong magnetic field, we find that generations of gapped Dirac electrons systematically reappear in Zak's magnetic miniband spectra, for rational values of the magnetic flux through the moire supercell. The fractal Hofstadter spectra, in the vicinity of such flux, can be described in terms of Landau levels, traced to the recurrent gapped Dirac electrons. Since this Landau level spectrum contains a zeroth energy level, separated by the largest gap from the rest of the spectrum, this determines a specific hierarchy of minigaps in the Hofstadter butterfly, and a peculiar sequence of dominant incompressible electron states. By studying semiconductor materials databases, one finds that there are also several crystals with hexagonal facets almost commensurate with the "the square root of"3 x "the square root of"3 Kekule lattice of graphene: InAs(lll)B, InP(lll)B, PdTe2, PtTe2, InSe, hGaTe. Using generic phenomenological theory, for superlattice effects created by such substrates on the Dirac electrons in graphene, we find that a typical miniband spectrum has a band gap between the first two minibands. Separately, a theory of the electron-phonon coupling and Raman scattering by phonons in graphene is developed. Also, we systematically investigate the effect of moire superlattice perturbation on graphene's, otherwise featureless, optical absorption spectra.