  We provide a gentle introduction to arc spaces, motivic integration and
stringy invariants. We explain the basic concepts and first results, including
the p-adic number theoretic pre-history, and we provide concrete examples.
  The text is a slightly adapted version of the 'extended abstract' of the
author's talks at the 12th MSJ-IRI "Singularity Theory and Its Applications"
(2003) in Sapporo. At the end we included a list of various recent results.
