  It is shown that the main geometrical objects involved in all the symmetries
or supersymmetries of the Dirac operators in curved manifolds of arbitrary
dimensions are the Killing vectors and the Killing-Yano tensors of any ranks.
The general theory of external symmetry transformations associated to the usual
isometries is presented, pointing out that these leave the standard Dirac
equation invariant providing the correct spin parts of the group generators.
Furthermore, one analyzes the new type of symmetries generated by the
covariantly constant Killing-Yano tensors that realize certain square roots of
the metric tensor. Such a Killing-Yano tensor produces simultaneously a
Dirac-type operator and the generator of a one-parameter Lie group connecting
this operator with the standard Dirac one. In this way the Dirac operators are
related among themselves through continuous transformations associated to
specific discrete ones. It is shown that the groups of this continuous symmetry
can be only U(1) or SU(2), as those of the (hyper-)Kahler spaces, but arising
even in cases when the requirements for these special geometries are not
fulfilled. To exemplify, the Euclidean Taub-NUT space with its Dirac-type
operators is presented in much details, pointing out that there is an
infinite-loop superalgebra playing the role of a closed dynamical algebraic
structure. As a final topic, we go to consider the properties of the Dirac-type
operators of the Minkowski spacetime.
