The intuition behind the notion of a cycle-free partial order (CFPO) is that it should be a partial ordering (X, ≤ ) in which for any sequence of points (x0, x1;…, xn–1) with n ≤ 4 such that xi is comparable with xi+1 for each i (indices taken modulo n) there are i and j with j ╪ i, i + 1 such that xj lies between xi and xi+1. As its turn out however this fails to capture the intended class, and a more involved definition, in terms of the ‘Dedekind–MacNeille completion’ of X was given by Warren. An alternative definition involving the idea of a betweenness relation was proposed by P. M. Neumann . It is the purpose of this paper to clarify the connections between these definitions, and indeed between the ideas of semi-linear order (or ‘tree’), CFPO, and the betweenness relations described in . In addition I shall tackle the issue of the axiomatizability of the class of CFPOs.