The study of the maps between classifying spaces of compact Lie groups has been one of the highlights of algebraic topology in the final quarter of the XXth century.

From a simply connected compact connected Lie group we obtain a finite dimensional Lie algebra, and from a finite dimensional Lie algebra we obtain a Cartan matrix. A Cartan matrix A=(a_i,j) is a positive definite matrix with integer coefficients such that a_i,i=2, a_i,j ≥ 0 and a_i,j=0 implies a_j,i=0.
All this proces can be inverted and we can recover the Lie algebra from the Cartan matrix and ``integrate'' this Lie algebra to obtain a simply connected, compact, connected Lie group.

Consider now a generalizad Cartan matrix, that is a non necessarily positive definite Cartan matrix. We can construct an integrable Lie algebra (non finite dimensional, in general) and from it a topologicla group. The result of these constructions are the called Kac-Moody algebras and Kac-Moody groups.

From a homotopy point of view the Kac-Moody groups were studied by N. Kitchloo (cohomological properties) and this
results took us to consider other well known results in compact Lie groups to be generalized to Kac-Moody groups.

The main result of the thesis is the study of the mapping space [BK,BK], where K is a rank 2 Kac-Moody group.

In order to understand [BK,BK] we must calculate [BT,BK], where T is a maximal torus in K. Here we get results which do not agree with the compact Lie group case: there exist maps from BT to BK which do not come from representations.

With this study we get a complete description of the subspace of [BT,BK] which involve all the maps which come from maps of [BK,BK]. This classification will allow us to understand the space [BK,BK],  after proving that the map induced by the inclusion [BK,BK] in [BT,BK] is injective.

Studying this we get other results like the characterization of the homotopy type of the rank 2 Kac-Moody groups (in particular we obtain non-isomorphic Kac-Moody groups with the same classifying space) and a characterization of the possible degrees of maps from BK to BK.