  The quantum resonances (QRs) of the kicked particle are studied in a most
general framework by also considering {\em arbitrary} periodic kicking
potentials. It is shown that QR can arise, in general, for {\em any rational}
value of the Bloch quasimomentum. This is illustrated in the case of the main
QRs for arbitrary potentials. In this case, which is shown to be precisely
described by the linear kicked rotor, exact formulas are derived for the
diffusion coefficients determining the asymptotic evolution of the average
kinetic energy of either an incoherent mixture of plane waves or a general wave
packet. The momentum probability distribution is exactly calculated and studied
for a two-harmonic potential. It clearly exhibits new resonant values of the
quasimomentum and it is robust under small deviations from QR.
