We discuss the possible relevance of gravitational-wave (GW) experiments for physics at very high energy. We examine whether, from the experience gained with the computations of various specific relic GW backgrounds, we can extract statements and order of magnitude estimates that are as much as possible model-independent, and we try to distinguish between general conclusions and results related to specific cosmological mechanisms. We examine the statement that the Virgo/LIGO experiments probe the Universe at temperatures $T\sim 10^{7}-10^{10}$ GeV (or timescales $t\sim 10^{-20}-10^{-26}$ sec) and we consider the possibility that they could actually probe the Universe at much higher energy scales, including the typical scales of grand unification, string theory and quantum gravity. We consider possible scenarios, depending on how the inflationary paradigm is implemented. We discuss the prospects for detection with present and planned experiments. In particular, a second Virgo interferometer correlated with the planned one, and located within a few tens of kilometers from the first, could reach an interesting sensitivity for stochastic GWs of cosmological origin.

26 pages, Latex, 1 figure