  A central role in the description of topological phases of matter in
(2+1)-dimensions is played by the braid groups. Motions exchanging non-abelian
anyons in the disk induce unitary representations of the braid groups. It is of
interest to generalise these ideas to different manifolds, to particles which
are not point like (loop or string excitations in a 3-ball, for example) and to
motions of particles that do not necessarily start and end in the same
configuration. In this paper we construct a categorical framework for studying
such generalisations of anyons. Mathematically, there are a number of ways to
define the braid groups but the one that arises from the physical picture of
particles moving in space is the definition as (equivalence classes of) motions
of points in a disk. In this paper we construct for each manifold $M$ its
motion groupoid $Mot_M$. (One recovers the classical definitions of motion
groups associated to a pair of a manifold $M$ and a subset by considering the
automorphisms of the corresponding object.)
  Braid groups can be equivalently defined as the boundary fixing mapping class
groups of points in a disk. We also give a construction of a mapping class
groupoid $MCG_M$ associated to a manifold. It is not the case that for all
manifolds the mapping class groupoid $MCG_M$ and motion groupoids $Mot_M$ are
isomorphic. We give conditions under which we have an isomorphism (for
$M=[0,1]^n$ for any $n\in \mathbb{N}$ for example).
  We will investigate the constructions of motion groupoids and mapping class
groupoids, and their relations, in the topological category. We expect that our
constructions and results can be formulated in the smooth category.
