A permutation group is semiprimitive if each normal subgroup is transitive or semiregular. This class of groups generalises the classes of primitive and quasi primitive groups, and has attracted most interest due to problems in algebraic graph theory. In this talk, I will survey some recent joint work with Kyle Rosa and Cheryl Praeger where we explored the question of whether certain bounds (in terms of degree) that hold for primitive groups could be adapted to the class of semiprimitive groups. Motivated by "useful'' results for the class of primitive groups, we investigated bounds on order, base size and fixity. In general, we found that similar bounds hold, or, one can establish even stronger bounds if we exclude certain families of primitive groups.