The atomic hydrogen gas left over from the Big Bang was affected by processes ranging from quantum fluctuations during the early epoch of inflation to irradiation by the first galaxies at late times. Mapping this gas through its resonant 21cm line serves a dual role as a powerful probe of both fundamental physics and astrophysics. Current cosmological data sets (such as galaxy surveys or the microwave background) cover only 0.1% of the comoving volume of the observable Universe. 21cm observations hold the potential of mapping matter through most of the remaining volume. Radio observatories are currently being designed and constructed with this goal in mind. The three-dimensional 21cm maps could potentially set unprecedented statistical constraints on the power spectrum of cosmic density fluctuations and its gravitational growth with cosmic time. The reduced uncertainties could allow for precise measurements of fundamental parameters, such as the mass of the neutrino or the equation of state of the dark energy (from acoustic oscillations in the 21cm power spectrum), and will test generic predictions of cosmic inflation for deviations of the density fluctuations from scale invariance and gaussianity. The measured gravitational growth of the fluctuations with cosmic time would constrain the nature of the dark matter or alternative theories of gravity.