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In this expository article, we outline the theory of harmonic differentialforms and its consequences. We provide self-contained proofs of the followingimportant results in differential geometry: (1) Hodge theorem, which statesthat for a compact complex manifold, the de Rham cohomology group is isomorphicto the group of harmonic forms, (2) Hodge decomposition theorem, which statesthat for a K\"ahler manifold, the de Rham cohomology group decomposes into theDolbeault cohomology groups, and (3) The Kodaira Embedding theorem, which givesa criterion of when a compact complex manifold is in fact a smooth complexprojective variety. The basic theory of vector bundles is also contained forcompleteness.

Harmonic Forms, Hodge Theory and the Kodaira Embedding Theorem