The thesis centres around two problems in the enumeration of pgroups. Define f_{φ}(p^{m}) to be the number of (isomorphism classes of) groups of order p^{m} in an isoclinism class φ. We give bounds for this function as φ is fixed and m varies and as m is fixed and φ varies. In the course of obtaining these bounds, we prove the following result. We say a group is reduced if it has no nontrivial abelian direct factors. Then the rank of the centre Z(P) and the rank of the derived factor group PP' of a reduced pgroup P are bounded in terms of the orders of PZ(P)P' and P'∩Z(P). A long standing conjecture of Charles C. Sims states that the number of groups of order p^{m} is p^{2andfrasl;27m3+O(m2)}. (1) We show that the number of groups of nilpotency class at most 3 and order p^{m} satisfies (1). We prove a similar result concerning the number of graded Lie rings of order p^{m} generated by their first grading.
