The thesis centres around two problems in the enumeration of p-groups. Define fφ(pm) to be the number of (isomorphism classes of) groups of order pm 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 non-trivial abelian direct factors. Then the rank of the centre Z(P) and the rank of the derived factor group P|P' of a reduced p-group P are bounded in terms of the orders of P|Z(P)P' and P'∩Z(P). A long standing conjecture of Charles C. Sims states that the number of groups of order pm is
p2andfrasl;27m3+O(m2). (1) We show that the number of groups of nilpotency class at most 3 and order pm satisfies (1). We prove a similar result concerning the number of graded Lie rings of order pm generated by their first grading.