  This is a contracted version of a series of lectures for graduate and
undergraduate students given at the {\sl "VI Seminario Nazionale di Fisica
Teorica"} (Parma, September 1997), at the Second Int. Conference {\sl "Around
VIRGO"} (Pisa, September 1998), and at the Second {\sl SIGRAV} School on {\sl
"Gravitational Waves in Astrophysics, Cosmology and String Theory"} (Center "A.
Volta", Como, April 1999). The aim is to provide an elementary, self-contained
introduction to string cosmology and, in particular, to the background of relic
cosmic gravitons predicted in the context of the so-called "pre-big bang"
scenario. No special preparation is required besides a basic knowledge of
general relativity and of standard (inflationary) cosmology. All the essential
computations are reported in full details either in the main text or in the
Appendices. For a deeper and more complete approach to the pre-big bang
scenario the interested reader is referred to the updated collection of papers
available at {\tt http://www.to.infn.it/\~{}gasperin/}.
