Although many abstractions of hemisystems of H (3, q 2) have been introduced since the first infinite family was discovered by Cossidente and Penttila in 2005, none have been applied to relative hemisystems. This is the main focus of this dissertation. We prove that every nontrivial relative m-cover is a relative hemisystem. We also give the first examples of relative hemisystems on nonclassical generalised quadrangles and study other generalisations in the classical case. Finally, we present computational results on the nonexistence of (q+ 1) 12-ovoids of DH (5, q 2) for q= 3 and 5, using a simplification of the problem