  Since the foundational work of Chenciner and Montgomery in 2000 there has
been a great deal of interest in choreographic solutions of the n-body problem:
periodic motions where the n bodies all follow one another at regular intervals
along a closed path. The principal approach combines variational methods with
symmetry properties. In this paper, we give a systematic treatment of the
symmetry aspect. In the first part we classify all possible symmetry groups of
planar n-body, collision-free choreographies. These symmetry groups fall in to
2 infinite families and, if n is odd, three exceptional groups. In the second
part we develop the equivariant fundamental group and use it to determine the
topology of the space of loops with a given symmetry, which we show is related
to certain cosets of the pure braid group in the full braid group, and to
centralizers of elements of the corresponding coset. In particular, we refine
the symmetry classification by classifying the connected components of the set
of loops with any given symmetry. This leads to the existence of many new
choreographies in $n$-body systems governed by a strong force potential.
