We present an extensive density functional theory (DFT) based investigation of the electronic structures of ABC-stacked N-layer graphene. It is found that for such systems the dispersion relations of the highest valence and the lowest conduction bands near the K point in the Brillouin zone are characterised by a mixture of cubic, parabolic, and linear behaviours. When the number of graphene layers is increased to more than three, the separation between the valence and conduction bands decreases up until they touch each other. For five and six layer samples these bands show flat behaviour close to the K point. We note that all states in the vicinity of the Fermi energy are surface states originated from the top and/or bottom surface of all the systems considered. For the trilayer system, N=3, pronounced trigonal warping of the bands slightly above the Fermi level is directly obtained from DFT calculations.