We tell the history of Moufang
generalized quadrangles. We review some Moufang-like conditions, and classify finite quasi-transitive
generalized quadrangles. These are
generalized quadrangles so that for any two non-concurrent lines U,V of the GQ, and for some W{U,V}
, the group of
generalized homologies with axes U and V acts transitively on the points of W incident with neither U nor V.
As a by-product of the proof, we will show, without the classification of finite simple groups, that a finite generalized quadrangle that admits a BN-pair of rank 2 is classical or dual classical if and only if it admits at least one nontrivial homology (!).