This work is divided into three parts, all of which are concerned with the characterisation of certain families of classical groups as doubly transitive permutation groups satisfying certain extra hypotheses. The first part is purely expository. It culminates in the characterisation andndash; due to Marggraf andndash; of the affine groups over GF(2). The second part deals with a characterisation of certain collineation groups of protective spaces as Jordan groups which admit a Jordan set of prime power cardinality and which have extra restrictions on some Sylow subgroups. The third part consists of results obtained while attempting to establish that insoluble groups of prime degree p (andgt;7), whose Sylow p-subgroups have index 3 in their normalisers, are of the form PSL(3,q), for suitable prime powers q.