We formulate a continuum model to study the low-energy electronic structure of heterostructures formed by graphene on a strong three-dimensional topological insulator (TI) for the case of both commensurate and incommensurate stacking. The incommensurability can be due to a twist angle between graphene and the TI surface or a lattice mismatch between the two systems. We find that the proximity of the TI induces in graphene a strong enhancement of the spin-orbit coupling that can be tuned via the twist angle.The surfaces of strong three-dimensional (3D) topological insulators (TIs) [1] and graphene [2,3] have very similar low-energy electronic structures: the conduction and valence bands touch at isolated points, the Dirac points (DPs), and around these points the fermionic excitations are well described as massless two-dimensional (2D) chiral Dirac fermions for which the phase of a twostate quantum degree of freedom is locked with the momentum direction. However, there are also qualitative differences: (i) in graphene the chirality is associated with the sublattice degree of freedom whereas in a TI surface (TIS) it is associated with the electron spin; (ii) in graphene the number of DPs is even whereas in a TIS it is odd ; (iii) in TIs the electron-phonon scattering is much stronger than that in graphene. Therefore, the transport properties of graphene [4] and TIs are different in significant aspects: in graphene, because the intrinsic spin-orbit (SO) coupling is negligible [5][6][7][8], no quantum spin Hall effect is expected, contrary to the case in a TI; graphene has the highest room-temperature mobility, whereas TIs have very low mobilities. These facts, together with the recent experimental progress [9], motivated us to study graphene-TI heterostructures, in which the proximity to a TI is expected to enhance the SO coupling of graphene and create a novel 2D system with nontrivial spin textures and high, room-temperature, electron mobility. This approach to enhance the SO coupling in graphene appears to be more practical than previously proposed approaches [10] that rely on doping graphene with heavy adatoms.In this work, we study the low-energy electronic structure of heterostructures formed by one sheet of graphene placed on the conducting surface of a 3D TI. Our results show that not only can the proximity of a TIS enhance the SO coupling in graphene and bilayer graphene (BLG), but also that this enhancement can be controlled via the relative rotation, the twist angle, between the graphene lattice and the TI's lattice. The presence of a relative rotation, in general, induces an incommensurate stacking of the graphene and the substrate [11] [12-36]. As a consequence we develop and present a theory that is able to take into account the incommensurability between graphene and the TIS. This cannot be achieved via standard approaches, such as density functional theory [37], and tight-binding models, due to the computational cost of these approaches for incommensurate structures. A continuous model, on the other...