  Transports preserving the angle between two contravariant vector fields but
changing their lengths proportional to their own lengths are introduced as
''conformal'' transports and investigated over spaces with contravariant and
covariant affine connections (whose components differ not only by sign) and
metrics. They are more general than the Fermi-Walker transports. In an
analogous way as in the case of Fermi-Walker transports a conformal covariant
differential operator and its conformal derivative are defined and considered
over the above mentioned spaces. Different special types of conformal
transports are determined inducing also Fermi-Walker transports for orthogonal
vector fields as special cases. Conditions under which the length of a non-null
contravariant vector field could swing as a homogeneous harmonic oscillator are
established. The results obtained regardless of any concrete field
(gravitational) theory could have direct applications in such types of
theories.
  PACS numbers: 04.90.+e; 04.50.+h; 12.10.Gq; 02.40.Vh
