Recently, M. Bestvina and N. Brady have exhibited groups that are of type FP but not finitely presented. We give explicit presentations for groups of the type considered by Bestvina-Brady. This leads to algebraic proofs of some of their results. Let ∆ be a finite flag complex, that is, a finite simplicial complex that contains a simplex bounding every complete subgraph of its 1-skeleton. The associated rightangled Artin group G∆ is the group given by the presentation with generators the vertex set of ∆, and relators the commutators [v, w] for each pair of adjacent vertices in ∆. For example, the n-simplex corresponds to a free abelian group of rank n+1, a complex consisting of n points corresponds to the free group Fn of rank n, the group corresponding to the square is (F2), and the group corresponding to the octahedron is (F2). Provided that ∆ is non-empty, there is a homomorphism from G∆ onto the integers, that takes every generator to 1. The group H∆ is defined to be the kernel of this homomorphism. Remarkable recent work of Mladen Bestvina and Noel Brady has shown that the homological finiteness properties of the group H∆ are controlled by the topology of the complex ∆ . They show that H∆ is finitely generated if and only if ∆ is connected, H∆ is finitely presented if and only if ∆ is 1-connected, and H∆ is of type FP (n) if and only if ∆ is (n−1)-acyclic. Precursors of this result include J. Stallings’ group that is finitely presented but not of type FP (3), which is H∆ in the case when ∆ is the octahedron , and R. Bieri’s group of type FP (n) but not of type FP (n + 1), which is H∆ in the case when ∆ is a join of (n+ 1) pairs of points (section 2.6 of ). The arguments used by Bestvina and Brady are geometric, and they do not give presentations for the groups that they consider. Theorem 1 of this paper gives a presentation for H∆ for any connected ∆. The generators in the presentation are the edges of ∆, and each 1-cycle in ∆ gives rise to an infinite family of relators. In the case when ∆ is simply connected, it is shown how to reduce this presentation to a finite one. This gives an independent and purely algebraic proof that H∆ is finitely presented when ∆ is simply connected. It would be interesting to give a similar proof of the converse. In Proposition 4 and Corollary 5, we review some Received by the editors May 17, 1997. 1991 Mathematics Subject Classification. Primary 20F36; Secondary 20E07, 20F32.