The groupoid identity x(xy) = y appears in dening several classes of groupoids, such as Steiner's loops which are closely related to Steiner's triple systems, the class of can- cellative groupoids with property (2, 5), Boolean groups, and groupoids which exhibit orthogonality of quasigroups. Its dual identity is one of the dening identities for the variety of quasigroups corresponding to strongly 2-perfect m-cycle systems. In this paper we consider the following varieties of groupoids: V = V ar(x(xy) = y), Vc = V ar(x(xy) = y, xy = yx), Vu = V ar(x(xy) = y, (xy)y = xy), Vi = V ar(x(xy) = y, (xy)y = yx). Suitable canonical constructions of free objects in each of these varieties are given and several other structural properties are presented. Some problems of enumeration of groupoids are also resolved. It is shown that each Vi -groupoid denes a Steiner quintuple system and vice versa, implying existence of Steiner quintuple systems of enough large nite cardinality.