The study of three-manifolds via their Heegaard splittings was initiated by Poul Heegaard in 1898 in his thesis. Our approach to the subject is through almost normal surfaces, as introduced by Hyam Rubinstein [Geo. Top. (Athens), 1993] and distance, as introduced by John Hempel [Topology, 2001].

Among the results presented is a proof that every closed, orientable three-manifold has only finitely many Heegaard splittings with distance greater than 4, a new recognition algorithm for surface bundles over the circle, and a series of results which bound the distance of a splitting in terms of its structure as an almost normal surface.