  We show that the transmission through single and double {\delta}-function
potential barriers of strength P in bilayer graphene is periodic in P with
period {\pi}. For a certain range of P values we find states that are bound to
the potential barrier and that run along the potential barrier. Similar
periodic behaviour is found for the conductance. The spectrum of a periodic
succession of {\delta}-function barriers (Kronig-Penney model) in bilayer
graphene is periodic in P with period 2{\pi}. For P smaller than a critical
value, the spectrum exhibits two Dirac points while for P larger than this
value an energy gap opens. These results are extended to the case of a
superlattice of {\delta}-function barriers with P alternating in sign between
successive barriers; the corresponding spectrum is periodic in P with period
{\pi}.
