  The Corlette-Donaldson-Hitchin-Simpson's correspondence states that, on a
compact K\"ahler manifold $(X, \omega )$, there is a one-to-one correspondence
between the moduli space of semisimple flat complex vector bundles and the
moduli space of poly-stable Higgs bundles with vanishing Chern numbers. In this
paper, we extend this correspondence to the projectively flat bundles case. We
prove that there is an equivalence of categories between the category of
$\omega$-semi-stable (poly-stable) Higgs bundles $(E, \overline{\partial}_{E},
\phi )$ with $(2rc_{2}(E)-(r-1)c_{1}^{2}(E))\cdot [\omega]^{n-2}=0 $ and the
category of (semi-simple) projectively flat bundles $(E, D)$ with
$\sqrt{-1}F_{D}=\alpha \otimes \mbox{Id}_{E}$ for some real (1,1)-form
$\alpha$. Furthermore, we also establish the above correspondence on some
compact non-K\"ahler manifolds. As its application, we obtain a vanishing
theorem of characteristic classes of projectively flat bundles.
