  We consider a distributed system of n identical mobile robots operating in
the two dimensional Euclidian plane. As in the previous studies, we consider
the robots to be anonymous, oblivious, dis-oriented, and without any
communication capabilities, operating based on the Look-Compute-Move model
where the next location of a robot depends only on its view of the current
configuration. Even in this seemingly weak model, most formation problems which
require constructing specific configurations, can be solved quite easily when
the robots are fully synchronized with each other. In this paper we introduce
and study a new class of problems which, unlike the formation problems so far,
cannot always be solved even in the fully synchronous model with atomic and
rigid moves. This class of problems requires the robots to permute their
locations in the plane. In particular, we are interested in implementing two
special types of permutations -- permutations without any fixed points and
permutations of order $n$. The former (called MOVE-ALL) requires each robot to
visit at least two of the initial locations, while the latter (called
VISIT-ALL) requires every robot to visit each of the initial locations in a
periodic manner. We provide a characterization of the solvability of these
problems, showing the main challenges in solving this class of problems for
mobile robots. We also provide algorithms for the feasible cases, in particular
distinguishing between one-step algorithms (where each configuration must be a
permutation of the original configuration) and multi-step algorithms (which
allow intermediate configurations). These results open a new research direction
in mobile distributed robotics which has not been investigated before.
