Graphene Moiré superlattice, a unique 2D periodical structure originated from the interaction between graphene and its supporting substrate h-BN, has attracted great interest recently. Employing epitaxial graphene on h-BN single crystals, we have investigated systematically the physical properties related to the Moiré superlattice. From transport measurements, we can observe the superlattice Dirac points at both electron side and hole side. Similar to the Dirac point, the superlattice Dirac points have insulator behaviors. Under the action of magnetic field, the quantum Hall effects both in monolayer and bilayer graphenes are observed. Also, the Moiré superlattice can lead to the formation of self-similar mini-bands from the Landau fan diagram. According to the infrared optical spectroscopy measurements, the transitions between different Landau levels are characterized by massive Dirac fermions and thus reveal a band-gap of ～38 meV. Moreover, without magnetic fields, an optical conductivity peak related to the Moiré superlattice appears. We use three spinor potential components to explain the optical conductivity peak and demonstrate that the pseudospin-mixing component plays a dominant role in the spinor potential. In addition, the spinor potential depends sensitively on the gate voltage, indicating that the electron–electron interactions play an important part in the renormalization of the spinor potential.