  We present experimental studies of the geometry-specific quantum scattering
in microwave billiards of a given shape. We perform full quantum mechanical
scattering calculations and find an excellent agreement with the experimental
results. We also carry out the semiclassical calculations where the conductance
is given as a sum of all classical trajectories between the leads, each of them
carrying the quantum-mechanical phase. We unambiguously demonstrate that the
characteristic frequencies of the oscillations in the transmission and
reflection amplitudes are related to the length distribution of the classical
trajectories between the leads, whereas the frequencies of the probabilities
can be understood in terms of the length difference distribution in the pairs
of classical trajectories. We also discuss the effect of non-classical "ghost"
trajectories that include classically forbidden reflection off the lead mouths.
