We present the extended GALEX Arecibo SDSS Survey (xGASS), a gas fraction-limited census of the atomic hydrogen (HI) gas content of 1179 galaxies selected only by stellar mass (M*=10^9^-10^11.5^M_{sun}_) and redshift (0.01<z<0.05). This includes new Arecibo observations of 208 galaxies, for which we release catalogues and HI spectra. In addition to extending the GASS HI scaling relations by one decade in stellar mass, we quantify total (atomic+molecular) cold gas fractions and molecular-to-atomic gas mass ratios, Rmol, for the subset of 477 galaxies observed with the IRAM 30m telescope. We find that atomic gas fractions keep increasing with decreasing stellar mass, with no sign of a plateau down to logM*/M_{sun}_=9. Total gas reservoirs remain HI-dominated across our full stellar mass range, hence total gas fraction scaling relations closely resemble atomic ones, but with a scatter that strongly correlates with Rmol, especially at fixed specific star formation rate. On average, Rmol weakly increases with stellar mass and stellar surface density {mu}*, but individual values vary by almost two orders of magnitude at fixed M* or {mu}*. We show that, for galaxies on the star-forming sequence, variations of Rmol are mostly driven by changes of the HI reservoirs, with a clear dependence on {mu}*. Establishing if galaxy mass or structure plays the most important role in regulating the cold gas content of galaxies requires an accurate separation of bulge and disc components for the study of gas scaling relations