CLASSICAL HODGE THEORY. Spectral Sequences and Mixed Hodge Structures. Complex Manifolds, Vector Bundles, Differential Forms. Sheaves and Cohomology. Harmonic Forms on Hermitian Manifolds. Hodge Theory on Compact Kählerian Manifolds. The Theory of Residues on a Smooth Divisor. Complex Spaces. DIFFERENTIAL FORMS ON COMPLEX SPACES. The Basic Example. Differential Forms in Complex Spaces. Mixed Hodge Structures on Compact Spaces. MIXED HODGE STRUCTURES ON NONCOMPACT SPACES. Residues and Hodge Mixed Structures: Leray Theory. Residues and Mixed Hodge Structures on Noncompact Manifolds. Mixed Hodge Structures in Noncompact Spaces: The Basic Example. Mixed Hodge Structures on Noncompact Singular Spaces.