TITLE Periodic orbits of the planar $N$--body problem with equal masses and all bodies on the same path AUTHOR Carles Sim\'o Departament de Matem\`atica Aplicada i An\`alisi Universitat de Barcelona Gran Via de les Corts Catalanes, 585 08007 Barcelona e-mail:carles@maia.ub.es To appear in ABSTRACT: Very few things are known about the solutions of the $N$--body problem, either under the action of the Newtonian potential or other homogeneous potentials of the form $1/r^a,\, a>0.$ Only some partial results about central configurations are available. In this lecture we review some recent results concerning the existence of periodic solutions of the planar $N$--body problem, with all masses equal, such that the $N$ bodies travel along the some path on the plane. These orbits are denoted as {\em choreographies}. A huge amount of families have been found numerically. Existence proofs are based on variational methods and require, up to now, the exponent $a\ge 2$ in the potential.