Using the Kirillov orbit method, novel methods from p-adic integration and Clifford theory, we study representation zeta functions associated to compact p-adic analytic groups. In particular, we give general estimates for the abscissae of convergence of such zeta functions.We compute explicit formulae for the representation zeta functions of some compact p-adic analytic groups, defined over a compact discrete valuation ring o of characteristic 0. These include principal congruence subgroups of SL2(o), without any restrictions on the residue field characteristic of o, as well as the norm one group SL1(D) of a nonsplit quaternion algebra D over the field of fractions of o and its principal congruence subgroups. We also determine the representation zeta functions of principal congruence subgroups of SL3(o) in the case that o has residue field characteristic 3 and is unramified over Z3.Theorem 1.4. Let o be a compact discrete valuation ring of characteristic 0, with residue field of cardinality q and characteristic 3. Suppose that o is unramified over Z 3 . Then, for all m ∈ N, one hasThe four theorems are proved and discussed in Sections 2.3, 3.1, 4.2 and 5 respectively. Whilst all of the results in the current paper are of a local nature, they might be best appreciated in the 'global' context of representation zeta functions of arithmetic groups, as explained to some extent below and exposed, in much greater detail, in [2,3,4].