We study the scattering of massless scalar waves by a Kerr black hole by le
tting plane monochromatic waves impinge on the black hole. We calculate the
relevant scattering phase-shifts using the Prufer phase-function method, w
hich is computationally efficient and also reliable for high frequencies an
d/or large values of the angular multipole indices (l, m), We use the obtai
ned phase-shifts and the partial-wave approach to determine differential cr
oss sections and deflection functions. Results for off-axis scattering (wav
es incident along directions misaligned with the black hole's rotation axis
) are obtained for the first time. Inspection of the off-axis deflection fu
nctions reveals the same scattering phenomena as in Schwarzschild scatterin
g. In particular, the cross sections are dominated by the glory effect and
the forward (Coulomb) divergence due to the long-range nature of the gravit
ational field. In the rotating case the overall diffraction pattern is 'fra
me-dragged' and as a result the glory maximum is not observed in the exact
backward direction. We discuss the physical reason for this behaviour, and
explain it in terms of the distinction between prograde and retrograde moti
on in the Kerr gravitational field. Finally, we also discuss the possible i
nfluence of the so-called superradiance effect on the scattered waves.