Using a general symmetry-based approach, we provide a classification of generic miniband structures for electrons in graphene placed on substrates with the hexagonal Bravais symmetry. In particular, we identify conditions at which the first moiré miniband is separated from the rest of the spectrum by either one or a group of three isolated mini Dirac points and is not obscured by dispersion surfaces coming from other minibands. In such cases the Hall coefficient exhibits two distinct alternations of its sign as a function of charge carrier density.PACS numbers: 73.22.Pr,73.21.Cd, Recently, it has been demonstrated that the electronic quality of graphene-based devices can be dramatically improved by placing graphene on an atomically flat crystal surface, such as hexagonal boron nitride (hBN) [1][2][3][4][5][6][7]. At the same time, graphene's electronic spectrum also becomes modified, acquiring a complex, energydependent form caused by incommensurability between the graphene and substrate crystal lattices [8][9][10]. In particular, for graphene placed on hBN, the difference between their lattice constants and crystallographic misalignment generate a hexagonal periodic structure known as a moiré pattern [2,3,[8][9][10]. The resulting periodic perturbation, usually referred to as a superlattice, acts on graphene's charge carriers and leads to multiple minibands and the generation of secondary Dirac-like spectra. The resulting new Dirac fermions present yet another case where graphene allows mimicking of QED phenomena under conditions that cannot be achieved in particle physics experiments. In contrast to relativistic particles in free space, the properties of secondary Dirac fermions in graphene can be affected by a periodic sublattice symmetry breaking and modulation of carbon-carbon hopping amplitudes, in addition to a simple potential modulation. The combination of different features in the modulation results in a multiplicity of possible outcomes for the moiré miniband spectrum in graphene which we systematically investigate in this article.To describe the effect of a substrate on electrons in graphene at a distance, d, much larger than the spacing, a, between carbon atoms in graphene's honeycomb lattice, we use the earlier observation [8][9][10][11][12][13][14] that, at d a the lateral variation of the wavefunctions of the p z carbon orbitals is smooth on the scale of a. This is manifested in the comparable sizes of the skew and vertical hopping in graphite and permits an elegant continuummodel description [11][12][13][14] of the interlayer coupling in twisted bilayers and the resulting band structure. A similar idea applied to graphene on a hBN substrate [8][9][10] suggests that a substrate perturbation for Dirac electrons in graphene can be described in terms of simple harmonic functions corresponding to the six smallest reciprocal lattice vectors of the moiré superlattice.Below, we shall use a similar approach to analyse the generic properties of moiré minibands for electrons in graphene subjected to a subs...