A unital is a 2-(q³+1,q+1,1)-design. Unitals embedded in finite projective planes are studied in two settings: the classical projective plane PG(2,q²) and the Hughes plane of order q². The main results involve a characterization of Buekenhout-Metz unitals in the former and the identification of a new family of unitals in the latter.

In the study of the Buekenhout-Metz unitals embedded in PG(2,q²⁾, the points of the unital which lie on the tangent lines through a point not belonging to the unital are classified. Moreover, a group theoretic characterization of Buekenhout-Metz unitals is obtained. It is shown that a unital U embedded in PG(2,q²) is a Buekenhout-Metz unital if and only if U admits a linear collineation group that is a semidirect product of a Sylow p-subgroup of order q³ by a subgroup of order q -1. In addition, a new group theoretic characterization of the classical or Hermitian unital is obtained.

Several tools involving incidence in the Hughes plane are developed in order to study unitals embedded therein. New results are obtained regarding the Rosati unital embedded in the Hughes plane. The orbit structure of the points of the Rosati unital under the group which leaves the unital invariant is determined. Furthermore, the points of the Rosati unital which lie on the tangent lines through a point off the unital are determined. The Rosati unital is found to be a member of a previously unknown family of unitals in the Hughes plane. The collineation group leaving the unitals invariant is determined, as well as the orbit structure under a certain point-stabilizer. This new family of unitals is sorted according to projective equivalence and the Rosati unital is found to be self-dual.