This Chapter brings the first part of the book to its conclusion, with the construction of the Floer homology groups of a homology 3-sphere, using instantons over a 4-dimensional tube. Most of the technical work has been done in the previous two Chapters, but there are three further topics which we have kept for this Chapter. The first, which we take up in Section 5.1, is a discussion of compactness properties of instanton moduli spaces over manifolds with tubular ends. These are crucially important in Floer's theory, but the proofs are straightforward applications of the basic results summarised in Chapter 2. The next topic is the orientation of the moduli spaces or, better, of orientation line bundles formed from virtual index bundles. The key point here is a simple extension of the additive formula of Proposition 3.8. The other technical topic is a discussion of suitable perturbations of the instanton equation, which are constructed in Section 5.5. For purposes of exposition we give the main idea of Floer's theory at the earliest possible stage in this Chapter by working modulo 2 (which avoids orientations) and making a general position assumption (which avoids perturbations). These two extra topics are then fitted on to give the general definition of the Floer groups, using SU(2) bundles over homology spheres. In the last Section 5.6 we discuss a straightforward extension of the theory to SO(3) connections.