In algebraic geometry, a homogeneous variety is an algebraic variety on which an algebraic group acts transitively. Homogeneous varieties over an algebraically closed field are quotient varieties where is an algebraic group and a subgroup scheme (for instance, an algebraic subgroup).
Such varieties are always smooth quasi-projective varieties.
Classical examples are flag varieties (when is semisimple and a parabolic subgroup), or more generally homogeneous spherical varieties. Severi-Brauer varieties are examples of homogeneous varieties over a field without any rational points.