ABSTRACT. If A and B are nontrivial topological groups, not both discrete, such that their free product A 11 B is a sequential space, then it is sequential of order oji.1. Introduction.In [15], Ordman and Smith-Thomas prove that if the free topological group on a nondiscrete space is sequential then it is sequential of order uji. In particular, this implies that free topological groups are not metrizable or even Fréchet spaces. Our main result is the analogue of this for free products of topological groups. More precisely we prove that if A and B are nontrivial topological groups not both discrete and their free product A II B is sequential, then it is sequential of order ui. This result is then extended to some amalgamated free products. Our theorem includes, as a special case, the result of [10]. En route we extend the Ordman and Smith-Thomas result to a number of other topologies on a free group including the Graev topology which is the finest locally invariant group topology. (See [12].) It should be mentioned, also, that O.dman and SmithThomas show that the condition of AII B being sequential is satisfied whenever A and B are sequential fc^-spaces.