A Klein surface with boundary of algebraic genus \mathfrak{p}\geq 2 , has at most 12(\mathfrak{p}-1) automorphisms. The groups attaining this upper bound are called M^{\ast} -groups, and the corresponding surfaces are said to have maximal symmetry. The M^{\ast} -groups are characterized by a partial presentation by generators and relators. The alternating groups A_{n} were proved to be M^{\ast} -groups when n\geq 168 by M. Conder. In this work we prove that A_{n} is an M^{\ast } -group if and only if n\geq 13 or n=5,10 . In addition, we describe topologically the surfaces with maximal symmetry having A_{n} as automorphism group, in terms of the partial presentation of the group. As an application we determine explicitly all such surfaces for n\leq 14 . Each finite group G acts as an automorphism group of several Klein surfaces. The minimal genus of these surfaces is called the real genus of the group, \rho(G) . If G is an M^{\ast} -group then \rho(G)=\frac{o(G)}{12}+1 . We end our work by calculating the real genus of the alternating groups which are not M^{\ast} -groups.