Abstract
Andreas Floer [93] associated with every integral homology 3-sphere eight finitely generated abelian groups In(Σ), n = 0,..., 7, which are now referred to as the (instanton) Floer homology. The Floer homology is an invariant of orientation preserving diffeomorphism. It is a refinement of the Casson invariant λ(Σ) in that λ(Σ) is half the Euler characteristic of I*(Σ). The definition of I*(Σ) relies heavily on gauge theory in dimensions three and four. In this chapter, we define I*(Σ), review its properties, and give examples of calculations. We also explain relations between I*(Σ) and the symplectic Floer homology, between I*(Σ) and Donaldson polynomials, and describe various extensions of I*(Σ).