IN (11) J. H. C. Whitehead introduced the theory of regular neighbourhoods, which has become a basic tool in combinatorial topology. We extend the theory in three ways. First we relativize the concept, and introduce the regular neighbourhood N of X mod Y in M, where X and Y are two compact polyhedra in the manifold M, satisfying a certain condition called link-collapsibility. We prove existence and uniqueness theorems. The idea is that N should be a neighbourhood of X— Y, but should avoid Y as much as possible. The notion is extremely useful in practice, and is illustrated by the following examples. We assume M to be closed for the examples. (i) If Y = 0 then N is a, regular neighbourhood of X. Therefore the relative theory is a generalization of the absolute theory. (ii) If X is a manifold with boundary Y, then the interior of X lies in the interior of N and the boundary of X lies in the boundary of N; in other words X is properly embedded in N. (iii) Let Ibe a cone and suppose that X n Y is contained in the base of the cone. Then N is a ball containing X— Y in its interior, In Y in its boundary, and Y — X in its exterior. The last example was used in ((12) Lemma 6), and was one of the examples which suggested the need for a relative theory. Other illustrations of the use are to be found in the proofs of Theorem 2, Corollary 8, and Lemmas 7, 8, and 9 below, in the proof of Theorem 3 of (4), and in forthcoming papers by us on isotopy. Secondly, Whitehead proved a uniqueness theorem that said that any two regular neighbourhoods were (piecewise linearly) homeomorphic. We strengthen this result by showing them to be isotopic, keeping a smaller regular neighbourhood fixed (Theorem 2). In fact they are ambient isotopic provided that they meet the boundary regularly (Theorem 3), which is always the case if M is unbounded. Thirdly, Whitehead wrote the theory in the combinatorial category, and we rewrite it in the polyhedral category. The difference is that the combinatorial category consists of simplicial complexes and piecewiselinear maps, whereas the polyhedral category consists of polyhedra and piecewise-linear maps. In this paper by a polyhedron we mean a topological