  [abridged] We present constraints on the nature of the first galaxies
selected at 350 microns. The sample includes galaxies discovered in the deepest
blank-field survey at 350 microns (in the Bootes Deep Field) and also later
serendipitous detections in the Lockman Hole. Spectral energy distribution
templates are fit to identified counterparts, and the sample is found to
comprise IR-luminous galaxies at 1<z<3 predominantly powered by star formation.
The first spectrum of a 350-micron-selected galaxy provides an additional
confirmation, showing prominent dust grain features typically associated with
star-forming galaxies.
  Compared to submillimeter galaxies selected at 850 and 1100 microns, galaxies
selected at 350 microns have a similar range of far-infrared color
temperatures. However, no 350-micron-selected sources are reliably detected at
850 or 1100 microns. Galaxies in our sample with redshifts 1<z<2 show a tight
correlation between the far- and mid-infrared flux densities, but galaxies at
higher redshifts show a large dispersion in their mid- to far-infrared colors.
  The 350-micron flux densities (15<S(350)<40 mJy) place these objects near the
Herschel/SPIRE 350-micron confusion threshold, with the lower limit on the star
formation rate density suggesting the bulk of the 350-micron contribution will
come from less luminous infrared sources and normal galaxies.
