We report on AGN with extremely soft X-ray spectra observed with ROSAT. From their optical emission lines these objects are classified as narrow-line Seyfert 1 galaxies (NLS1), almost all with extremely large Fe II/Hβ ratios and relatively narrow optical lines of hydrogen. Our results are based on a systematic ROSAT study of 46 NLS1. 32 of these are located in the fields of view of ROSAT pointings and 31 are detected above a 5σ limit. We find that NLS1 have generally steeper soft X-ray continuum slopes than normal Seyfert 1s, and there may exist an anticorrelation between 0.1-2.4keV continuum slope and the FWHM of the Hβ line. Objects with steep 0.1-2.4keV continuum slopes and Hβ FWHM>3000km/s are clearly discriminated against by nature. When simple power-law models are fit to the data, photon indices reach values up to about 5, much higher than is usually seen in Seyfert 1s. We discuss steep ROSAT spectra in light of soft X-ray excess and hard X-ray tail models. We do not find evidence for large neutral hydrogen column densities over the Galactic column, and thus barring very high dust-to-gas ratios simple obscuration by dust and cold gas along the line of sight cannot easily explain the narrow optical hydrogen Balmer lines of NLS1. Many NLS1 show rapid soft X-ray variability and thus significant electron scattering of their X-rays seems unlikely. We consider models for NLS1 where they are Seyfert 1s with extremal values of pole-on orientation, black hole mass and/or accretion rate, warm absorption and BLR thickness and confront these models with the known properties of NLS1. All simple models appear to have drawbacks, but models with smaller mass black holes and thicker BLRs show some promise. We suggest specific further tests of the models. Steep ROSAT spectra suggest that the EUV and X-ray spectral distributions of NLS1 may be somewhat different than those of normal Seyfert 1s, and these different spectral energy distributions can strongly influence BLR cloud formation and confinement. NLS1 may be analogous to the high ultrasoft states of Galactic black hole candidates.