In this paper, the classical Hankel transformation is extended to a certain space of generalized functions. We construct a space Tμ δ, B of testing functions, such that xJμ(xy) is in Tμ δ, B for every y and gt; 0. The generalized Hankel transformation hμf of fϵ Tμ, δ, B, the dual space of Tμ δ, B is defined by:(h´μf)(y)= and lt;f(x),xJμ (xy), y and gt; 0 .Several smoothness, boundedness and inversion theorems are proved. Moreover, Abelian theorems due to J.L. Griffith are extended to the space of distributions introduced. Finally, we analyze some applications of the generalized h'μ-transformation.