  A generic computation of a subset $A$ of $\mathbb{N}$ is a computation which
correctly computes most of the bits of $A$, but which potentially does not halt
on all inputs. The motivation for this concept is derived from complexity
theory, where it has been noticed that frequently, it is more important to know
how difficult a type of problem is in the general case than how difficult it is
in the worst case. When we study this concept from a recursion theoretic point
of view, to create a transitive relationship, we are forced to consider oracles
that sometimes fail to give answers when asked questions. Unfortunately, this
makes working in the generic degrees quite difficult. Indeed, we show that
generic reduction is $\mathbf\Pi^1_1-$complete. To help avoid this difficulty,
we work with the generic degrees of density-1 reals. We demonstrate how an
understanding of these degrees leads to a greater understanding of the overall
structure of the generic degrees, and we also use these density-1 sets to
provide a new a characterization of the hyperartithmetical Turing degrees.
