  This paper is a survey of recent work on the asymptotic behavior of the
syzygies of a smooth complex projective variety as the positivity of the
embedding line bundle grows.
  After a quick overview of results from the 1980s and 1990s concerning the
linearity of the first few terms of a resolution, we discuss a non-vanishing
theorem to the effect that from an asymptotic viewpoint, essentially all of the
syzygy modules that could be non-zero are in fact non-zero. We explain the
quick new proof of this result in the case of Veronese varieties due to Erman
and authors, and we explore some results and conjectures about the asymptotics
of Betti numbers. Finally we discuss the case of syzygies of weight one, and
the gonality conjecture on the syzygies of curves of large degree.
  The exposition also discusses numerous open questions and conjectures.
