The classical Fueter theorem addresses the fact that every holomorphic intrinsic func-tion of one complex variable induces a quaternionic monogenic function. Its higher dimensional generalizations to the Euclidean spaces ℝⁿ⁺¹, for n being odd and even, were respectively obtained by M. Sce (1957) and T. Qian (1997). Let f₀ be any holo-morphic intrinsic function of one complex variable defined on ℂ. The Fueter mapping is denoted by β(f₀)=(-∆)⁽ⁿ⁻¹⁾⁄²f₀, where f₀ is induced from f₀. We obtain that the Fueter mapping β is a surjection from the set of holomorphic intrinsic functions to the collection of axially monogenic functions. It contains three main theorems: the Fueter mapping axial form theorem, the Fueter mapping surjectivity theorem and the Fueter mapping monomial theorem. We also introduce the generalized Fueter mapping βₖ βₖ(f₀) :=(-Δ)ᵏ⁺⁽ⁿ⁻¹⁾⁄²(f₀(x)Pₖ(x)), where k can be all non-negative integers and Pₖ(x) is an inner spherical monogenic polynomial of degree k. Then we prove that for any f (the axially monogenic function of degree k defined on ℝⁿ⁺¹) there exists a holomorphic intrinsic function fₖ such that βₖ (fₖ)= f. Based on the result, We give a decomposition formula for the monogenic functions.