A class C of structures is said to be group universal if every group is the full automorphism group of some structure in C. In the present paper it is shown that each of the following classes is group universal: affine planes, projective planes, (m,n)-planes, (k,n)-Steiner systems, LP-spaces, covering geometries, commutative loops, quasigroups, commutative division algebras over fields and other related classes. In the proof the authors first select for a given group G a graph D with G=Aut(D). Then to each node and each edge of D a rigid structure is attached. Using free constructions, all these structures are amalgamated to a structure in C whose automorphism group is G. The proof uses geometric techniques as well as model-theoretic machinery.