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Introduction Part I. Preliminaries: 1. Holomorphic functions of many variables 2. Complex manifolds 3. Ka;hler metrics 4. Sheaves and cohomology Part II. The Hodge Decomposition: 5. Harmonic forms and cohomology 6. The case of Ka;hler manifolds 7. Hodge structures and polarisations 8. Holomorphic de Rham complexes Part III. Variations of Hodge Structure: 9. Families and deformations 10. Variations of Hodge structure Part IV. Cycles and Cycle Classes: 11. Hodge classes 12. The Abel-Jacobi map Bibliography Index.