Is it just me or In dimensions less than or equal to 3, any Riemannian metric of con- stant Ricci curvature has constant sectional curvature. Classical results in Riemannian geometry show that the universal cover of a closed manifold of constant positive curvature is diffeomorphic to the sphere and that the fun- damental group is identified with a finite subgroup of the orthogonal group acting linearly and freely on the universal cover. Thus, one can approach the Poincar ́e Conjecture and the more general 3-dimensional spherical space- form problem by asking the following question. Making the appropriate fundamental group assumptions on 3-manifold M, how does one establish the existence of a metric of constant Ricci curvature on M? The essential ingredient in producing such a metric is the Ricci flow equation introduced by Richard Hamilton in [29]: ∂g(t) = −2Ric(g(t)), ∂t where Ric(g(t)) is the Ricci curvature of the metric g(t). The fixed points (up to rescaling) of this equation are the Riemannian metrics of constant Ricci curvature. For a general introduction to the subject of the Ricci flow see Hamilton’s survey paper [34], the book by Chow-Knopf [13], or the book by Chow, Lu, and Ni [14]. The Ricci flow equation is a (weakly) para- bolic partial differential flow equation for Riemannian metrics on a smooth manifold. Following Hamilton, one defines a Ricci flow to be a family of Riemannian metrics g(t) on a fixed smooth manifold, parameterized by t in some interval, satisfying this equation. One considers t as time and studies the equation as an initial value problem: Beginning with any Riemann- ian manifold (M,g0) find a Ricci flow with (M,g0) as initial metric; that is to say find a one-parameter family (M,g(t)) of Riemannian manifolds with g(0) = g0 satisfying the Ricci flow equation. This equation is valid in all dimensions but we concentrate here on dimension 3. In a sentence, the method of proof is to begin with any Riemannian metric on the given smooth 3-manifold and flow it using the Ricci flow equation to obtain the constant curvature metric for which one is searching. There are two exam- ples where things work in exactly this way, both due to Hamilton. (i) If the initial metric has positive Ricci curvature, Hamilton proved over twenty years ago, [29], that under the Ricci flow the manifold shrinks to a point in finite time, that is to say, there is a finite-time singularity, and, as we approach the singular time, the diameter of the manifold tends to zero and the curvature blows up at every point. Hamilton went on to show that, in this case, rescaling by a time-dependent function so that the diameter is constant produces a one-parameter family of metrics converging smoothly to a metric of constant positive curvature. (ii) At the other extreme, in [36] Hamilton showed that if the Ricci flow exists for all time and if there is an appropriate curvature bound together with another geometric bound, then as t → ∞, after rescaling to have a fixed diameter, the metric converges to a metric of constant negative curvature. The results in the general case are much more complicated to formulate and much more difficult to establish. While Hamilton established that the Ricci flow equation has short-term existence properties, i.e., one can define g(t) for t in some interval [0,T) where T depends on the initial metric, it turns out that if the topology of the manifold is sufficiently complicated, say it is a non-trivial connected sum, then no matter what the initial metric is one must encounter finite-time singularities, forced by the topology. More seriously, even if the manifold has simple topology, beginning with an ar- bitrary metric one expects to (and cannot rule out the possibility that one will) encounter finite-time singularities in the Ricci flow. These singularities, unlike in the case of positive Ricci curvature, occur along proper subsets of the manifold, not the entire manifold. Thus, to derive the topological con- sequences stated above, it is not sufficient in general to stop the analysis the first time a singularity arises in the Ricci flow. One is led to study a more general evolution process called Ricci flow with surgery, first introduced by Hamilton in the context of four-manifolds, [35]. This evolution process is still parameterized by an interval in time, so that for each t in the interval of definition there is a compact Riemannian 3-manifold Mt. But there is a discrete set of times at which the manifolds and metrics undergo topolog- ical and metric discontinuities (surgeries). In each of the complementary intervals to the singular times, the evolution is the usual Ricci flow, though, because of the surgeries, the topological type of the manifold Mt changes as t moves from one complementary interval to the next. From an analytic point of view, the surgeries at the discontinuity times are introduced in order to ‘cut away’ a neighborhood of the singularities as they develop and insert by hand, in place of the ‘cut away’ regions, geometrically nice regions. This allows one to continue the Ricci flow (or more precisely, restart the Ricci flow with the new metric constructed at the discontinuity time). Of course, the surgery process also changes the topology. To be able to say anything useful topologically about such a process, one needs results about Ricci flow, and one also needs to control both the topology and the geometry of the surgery process at the singular times. For example, it is crucial for the topological applications that we do surgery along 2-spheres rather than sur- faces of higher genus. Surgery along 2-spheres produces the connected sum decomposition, which is well-understood topologically, while, for example, Dehn surgeries along tori can completely destroy the topology, changing any 3-manifold into any other. The change in topology turns out to be completely understandable and amazingly, the surgery processes produce exactly the topological operations needed to cut the manifold into pieces on which the Ricci flow can produce the metrics sufficiently controlled so that the topology can be recognized. The bulk of this book (Chapters 1-17 and the Appendix) concerns the establishment of the following long-time existence result for Ricci flow with surgery.