By using the tight binding approximation and Green function method, the electronic structure, density of state, electrical conductivity, heat capacity of BN and BN/graphene bilayers are investigated. The AA-, AB1- and AB2- BN/graphene bilayers have small gap unlike to BN bilayers which are wide band gap semiconductors. Unlike to BN bilayer, the energy gap of graphene/BN bilayers increases with external field. The magnitude of the change in the band gap of BN bilayers is much higher than the graphene/BN bilayers. Near absolute zero, the σ(T) is zero for BN bilayers and it increases with temperature until reaches maximum value then decreases. The BN/graphene bilayers have larger electrical conductivity larger than BN bilayers. For both bilayers, the specific heat capacity has a Schottky anomaly.
