This paper extends the Meijer transformation, M μ , given by ( M μ f ) ( p ) = 2 p Γ ( 1 + μ ) ∫ 0 ∞ f ( t ) ( p t ) μ / 2 K μ ( 2 p t ) d t , where f belongs to an appropriate function space, μ  ϵ  ( − 1 , ∞ ) and K μ is the modified Bessel function of third kind of order μ , to certain generalized functions. A testing space is constructed so as to contain the Kernel, ( p t ) μ / 2 K μ ( 2 p t ) , of the transformation. Some properties of the kernel, function space and its dual are derived. The generalized Meijer transform, M ¯ μ f , is now defined on the dual space. This transform is shown to be analytic and an inversion theorem, in the distributional sense, is established.