  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/H-beta ratios
and relatively narrow optical lines of hydrogen. Our results are based on a
systematic study of 46 NLS1. We find that NLS1 have generally steeper soft
X-ray continuum slopes than normal Seyfert 1s, and there may exist an
anticorrelation between 0.1-2.4 keV continuum slope and the FWHM of the H-beta
line. Objects with steep 0.1-2.4 keV continuum slopes and H-beta FWHM > 3000 km
s^{-1} 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 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.
