There are uncountably many sets that are not recursively enumerable, and the investigation of the Turing degrees of all sets is as central in recursion theory as the investigation of the recursively enumerable Turing degrees. Many degrees with special properties were constructed: "hyperimmune-free degrees" where every function computable relative to that degree is majorized by a (unrelativized) computable function; "high degrees" relative to which one can compute a function "f" which dominates every computable function "g" in the sense that there is a constant "c" depending on "g" such that "g(x) < f(x)" for all "x > c"; "random degrees" containing algorithmically random sets; "1-generic" degrees of 1-generic sets; and the degrees below the halting problem of limit-recursive sets.