Quasigroups form a variety only if two additional operations, namely \(/\) and \(\backslash\) are considered. Birkhoff formulated six identities that define equationally the variety of quasigroups. Evans proved that only the most natural four identities are needed. In this article the authors study which other four-tuples of Birkhoff's identities suffice to define quasigroups. Among all combinations, nine define quasigroups, four define larger classes and two remain open.