We define a real to be generically computable if there is a partial recursive function which correctly computes almost all of the bits of the real without necessarily halting on all inputs, but also without giving incorrect answers on any inputs. Our definition is originally motivated by an empirically observed phenomenon in complexity theory: that certain problems are much easier to solve in practice than would be suggested by their complexity classification, however, we study generic computability from a purely recursion theoretic point of view, primarily to investigate and understand the strange properties that this notion has. Relativized generic computation is not transitive, and thus not a true reduction procedure. There are no minimal pairs (or even finite minimal sets) for generic computation. We briefly sketch some of the new obstacles and new tools that are present in this context. If we change our definition of relativized generic computation to make it transitive, then the equivalence classes are each uncountable, and the relationship becomes Pi^1_1-complete. The Turing degrees can be shown to embed in the generic degrees, and under certain hypotheses, the generic degrees can be classified in terms of their relationships with the embedded Turing degrees.