  This is an introduction to the subject of the differential topology of the
space of smooth loops in a finite dimensional manifold. It began as the
background notes to a series of seminars given at NTNU and subsequently at
Sheffield. I am posting them in the hope that they will be useful to people
wishing to know a little about the basics of the subject. I have tried to make
them readable by anyone with a good grounding in finite dimensional
differential topology. Any comments and suggestions for improvement will be
very welcome.
  The topics covered are: the smooth structure of the space of smooth loops;
constructions involving vector bundles; submanifolds and tubular
neighbourhoods; and a short introduction to the geometry and semi-infinite
structure of loop spaces.
