  We consider an electrically charged particle on the Euclidean plane subjected
to a perpendicular magnetic field which depends only on one of the two
Cartesian co-ordinates. For such a ``unidirectionally constant'' magnetic field
(UMF), which otherwise may be random or not, we prove certain spectral and
transport properties associated with the corresponding one-particle
Schroedinger operator (without scalar potential) by analysing its ``energy-band
structure''. In particular, for an ergodic random UMF we provide conditions
which ensure that the operator's entire spectrum is almost surely absolutely
continuous. This implies that, along the direction in which the random UMF is
constant, the quantum-mechanical motion is almost surely ballistic, while in
the perpendicular direction in the plane one has dynamical localisation. The
conditions are verified, for example, for Gaussian and Poissonian random UMF's
with non-zero mean-values. These results may be viewed as ``random analogues''
of results first obtained by A. Iwatsuka [Publ. RIMS, Kyoto Univ. 21 (1985)
385] and (non-rigorously) by J. E. Mueller [Phys. Rev. Lett. 68 (1992) 385].
