The classical Kobayashi correspondence relates polystability of holomorphic bundles on Kähler manifolds and the existence of the Hermitean-Einstein metric. Hitchin considered some additional structures (Higgs fields) and proved the correspondence between the corresponding stability and the existence of Hermitean-Einstein metrics for Higgs bundles on a compact Riemann surface. The work of C. Simpson generalized it to smooth irreducible projective varieties. This book contains a discussion of the generalization of these results to the quasiprojective case. Results include an inequality for μL-stable parabolic Higgs bundles of the Bogomolov-Gieseker type, a deformation of any local system on a smooth quasiprojective variety to a variation of polarized Hodge structure, and a discussion of possible split quotients of the fundamental group of a smooth irreducible quasiprojective variety.