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Richard Thompson's famous group $F$ has the striking property that it can be realized as a dense subgroup of the group of all orientation-preserving homeomorphisms of the unit interval, but it can also be given by a simple 2-generator-2-relator presentation, in fact as the fundamental group of an aspherical complex with only two cells in each dimension. This monograph studies a natural generalization of $F$ that also includes Melanie Stein's generalized $F$-groups. The main aims of this monograph are the determination of isomorphisms among the generalized $F$-groups and the study of their...