  The Turing degree of a real measures the computational difficulty of
producing its binary expansion. Since Turing degrees are tailsets, it follows
from Kolmogorov's 0-1 law that for any property which may or may not be
satisfied by any given Turing degree, the satisfying class will either be of
Lebesgue measure 0 or 1, so long as it is measurable. So either the
\emph{typical} degree satisfies the property, or else the typical degree
satisfies its negation. Further, there is then some level of randomness
sufficient to ensure typicality in this regard. A similar analysis can be made
in terms of Baire category, where a standard form of genericity now plays the
role that randomness plays in the context of measure.
  We describe and prove a number of results in a programme of research which
aims to establish the properties of the typical Turing degree, where typicality
is gauged either in terms of Lebesgue measure or Baire category.
