In this talk Jon Magne Leinaas from University of Oslo reviews some of the basic ideas and questions related to the exchange symmetry of identical particles. This talks begins with discussing the braid description of particle exchange and some of the interesting question this raises with respect to geometry and topology. A particularly interesting question is whether the geometric understanding of particle exchange means that the spin-statistics theorem can be given a purely geometric formulation. Even if that so far has not been done, the correct relation between spin and statistics seems in some ways natural from the geometric point of view, as he will discuss. In the last part he will focus on the dynamics of systems of anyons in condensed matter systems, where the physical space has the character of a phase space rather than a configuration space. A semi-classical description is discussed, where the anyon parameter is coupled to an exclusion effect, which in the quantum description is known as exclusion statistics.