In this paper, we develop the theory of cellular folding of compact connected surfaces onto polygons. The first question that naturally arises from the definition of cellular foldings are. For a given compact connected surface M and a given polygon P_n, is there any cellular folding of M onto P_n?. Also if there are, are their topological types finitely many or infinitely many?. This is the existence problem.
We discuss this problem in the first part of this paper and we obtain a wide range of existence theorems for cellular foldings of a given surface onto a given polygon. Now, any simplicial folding decompose the surface into simplexes of dimensions 0, 1 and 2 which are called vertices, edges and faces respectively.
In the second part of this paper, we classify all the possible simplicial foldings of the sphere, the connected sum of n-tori and the connected sum of n-projective planes onto a polygon〖 P〗_3. For each surface we obtain certain relations satisfied by the number of vertices, edges and faces of the simplicial decomposition of the surface to get either regular simplicial folding or just a simplicial folding.