Compact and weakly compact elements of the group algebra L-1(G) of a locally compact group G, have been considered by a number of authors. In these investigations it has been shown that, if G is non-compact, then the only weakly compact element of L-1(G) is zero. Conversely, if G is compact, then every element of L-1(G) is compact. For 1 < p < infinity, let PMp (G) and PFp (G) denote the closure of L-1(G), considered as an algebra of convolution operators on L-p(G), with respect to the weak operator topology and the norm topology, respectively, in B((LP)-P-p(G)), the bounded linear operators on LP(G). We study the question of characterizing compact and weakly compact elements of the algebras PMp(G) and PFp(G).