In Memory of Dick and Brian Abstract A group is locally ﬁnite if every ﬁnite subset generates a ﬁnite subgroup. A group of linear transformations is ﬁnitary if each element minus the identity is an endomorphism of ﬁnite rank. The classiﬁcation and structure theory for locally ﬁnite simple groups splits naturally into two cases—those groups that can be faithfully represented as groups of ﬁnitary linear transformations and those groups that are not ﬁnitary linear. This paper completes the ﬁnitary case. We classify up to isomorphism those inﬁnite, locally ﬁnite, simple groups that are ﬁnitary linear but not linear. .