  We derive basic differential geometric formulae for surfaces in hyperbolic
space represented as envelopes of horospheres. The dual notion of parallel
hypersurfaces is also studied. The representation is applied to prove existence
and regularity theorems for Weingarten surfaces in H^3, which satisfy (1-a)K =
a(2-H), for an a < 0, and have a specified boundary curve at infinity. These
surfaces are shown to be closely connected to conformal mappings of domains in
S^2 into the unit disk and provide Riemannian interpretations for some
conformal invariants associated to such mappings.
  This paper was originally written in 1984, before I learned to use TeX, and
was typed by one of the secretaries in the Princeton Math Department. It was
more or less, my first original work after my dissertation. For some reason, I
was not able to get this paper published in a timely manner. The results and
perspective in this paper have proved to be useful to a variety of people, some
of whom asked me to render the article into TeX and post it to the arXiv. I had
been seriously thinking about doing this, when Martin Bridgemen sent me a
transcription of my original article into TeX. I am extremely grateful to him
for the effort he has put into this project.
  The paper is now formatted in a more or less modern AMS-article style, but
for lots of additional punctuation, a few corrections and some minor stylistic
changes, the content has been largely reproduced as it originally was. Remarks
about the 'state-of-the-art' in hyperbolic geometry are obviously way out of
date, as there has been enormous progress in many aspects of this still rich
subject.
