Abstract The Fueter mapping theorem is a fundamental result in quaternionic analysis relating slice hyperholomorphic functions and Fueter regular ones. The action of the Fueter map on quaternionic monomials leads to an interesting class of functions forming an Appell system with respect to the hypercomplex derivative. In this talk I will present two extensions of the Fueter map in the case of polyanalytic functions of a quaternionic variable. The first map is built upon a suitable global operator with non-constant coefficients allowing to construct Fueter regular functions starting from poly-slice hyperholomorphic ones. The second map allows to construct polyanalytic Fueter regular functions. Based on this second construction we introduce and study the main properties of a new family of Generalized-Appell polynomials which are poly-Fueter regular. I will discuss also how the polyanalytic Fueter maps act on a poly slice hyperholomorphic Bargmann transform. This gives arise to two integral transforms in the Fueter regular and polyanalytic Fueter regular setting.