Real generalized analytic functions are locally defined as sums of convergent generalized power series with coefficients in the real numbers; that is, power series whose exponents belong to a product of well-ordered sets of positive real numbers. In this talk we introduce generalized analytic manifolds and the blowing-up morphisms between them. We see that blowing-ups may exist or not depending on the existence of an ``standard analytic substructure''. We state several results of reduction of singularities for generalized analytic functions and we expose with detail the ``stratified version''. This is a work in collaboration with Jesús Palma-Márquez and Fernando Sanz-Sánchez.