  In general relativity black holes can be formed from regular initial data
that do not contain a black hole already. The space of regular initial data for
general relativity therefore splits naturally into two halves: data that form a
black hole in the evolution and data that do not. The spacetimes that are
evolved from initial data near the black hole threshold have many properties
that are mathematically analogous to a critical phase transition in statistical
mechanics.
  Solutions near the black hole threshold go through an intermediate attractor,
called the critical solution. The critical solution is either time-independent
(static) or scale-independent (self-similar). In the latter case, the final
black hole mass scales as $(p-p_*)^\gamma$ along any one-parameter family of
data with a regular parameter $p$ such that $p=p_*$ is the black hole threshold
in that family. The critical solution and the critical exponent $\gamma$ are
universal near the black hole threshold for a given type of matter.
  We show how the essence of these phenomena can be understood using dynamical
systems theory and dimensional analysis. We then review separately the analogy
with critical phase transitions in statistical mechanics, and aspects specific
to general relativity, such as spacetime singularities. We examine the evidence
that critical phenomena in gravitational collapse are generic, and give an
overview of their rich phenomenology.
