  Given a closed, oriented surface X of genus g>1, and a semisimple Lie group
G, let R_G be the moduli space of reductive representations of the fundamental
group of X in G. We determine the number of connected components of R_PGL(n,R),
for n>=4 even. In order to have a first division of connected components, we
first classify real projective bundles over such a surface. Then we achieve our
goal, using holomorphic methods through the theory of Higgs bundles over
compact Riemann surfaces. We also show that the complement of the Hitchin
component in R_SL(3,R) is homotopically equivalent to R_SO(3).
