  A complex contact threefold is a threefold with a two-dimensional
non-integrable holomorphic distribution. A contact curve on a contact threefold
is an integrable curve of the distribution. This work was inspired by two
papers of Bryant, in which he used complex contact geometry to study
superminimal surfaces in four-sphere and to investigate exotic holonomies. The
present paper is devoted to systematical studies of contact threefold and
contact curves on them. We generalize a result of Bryant and answer a question
of his.
