This thesis discusses upper bounds on the minimal number of elements d(G) required to generate a finite group G. We derive explicit upper bounds for the function d on transitive and minimally transitive permutation groups, in terms of their degree n. In the transitive case, bounds obtained first by Kovács and Newman, then by Bryant, Kovács and Robinson, and finally by Lucchini, Menegazzo and Morigi, show that d(G) = O(n/ √log n), for a transitive permutation group G of degree n. In this thesis, we find best possible estimates for the constant involved.