Suppose M is an oriented 3-manifold. A Dehn surgery on M (defined below) is a process by which M is altered by deleting a tubular neighbourhood of an embedded circle and replacing it again via some diffeomorphism of the boundary torus. It was shown by Lickorish [Li] and Wallace [Wa] that any closed oriented connected 3-manifold can be obtained from any other such manifold by a finite sequence of Dehn surgeries. Thus under this equivalence relation all closed oriented 3-manifolds are equivalent. We shall investigate this same question for more restricted classes of surgeries. In particular we shall insist that our Dehn surgeries preserve the integral (or rational) homology groups. Specifically, if M0 and M1 have isomorphic integral (respectively rational) homology groups, is there a sequence of Dehn surgeries, each of which preserves integral (respectively rational) homology, that transforms M0 to M1? What is the situation if we further restrict the Dehn surgeries to preserve more of the fundamental group? Is there a difference if we require ‘integral’ surgeries? We also show that these Dehn surgery relations are strongly connected to the following questions concerning another point of view towards understanding 3-manifolds. Is there a Heegard splitting of M0, M0 = H1 ∪fH2 (Hi are handlebodies of genus g and f is a homeomorphism of their common boundary surface), and a homeomorphism g of ∂H1 such that M1 has a Heegard splitting using g ∘ f as the identification? Since there are many natural subgroups of the mapping class group, such as the Torelli subgroup and the ‘Johnson subgroup’, one can ask the same question where g is restricted to lie in one of these subgroups. This is related to work of Morita on Casson's invariant for homology 3-spheres [Mo1]. Even under these restrictions it has been known for some time that any homology 3-sphere is related to S3. This fact has been used to define, calculate and understand invariants of homology 3-spheres (such as Casson's invariant) by choosing such a ‘path to S3’ in the ‘space’ of 3-manifolds.