Let X and Y be Banach spaces. A set M ? W(X, Y) (the space of all weakly compact operators from X into Y) is weakly equicompact if, for every bounded sequence (x n ) in X, there exists a subsequence (x k(n)) so that (Tx k(n)) is uniformly weakly convergent for T M. In this paper, the notion of weakly equicompact set is used to obtain characterizations of spaces X such that X ?1, of spaces X such that B X* is weak*sequentially compact and also to obtain several results concerning to the weak operator and the strong operator topologies. As another application of weak equicompactness, we conclude a characterization of relatively compact sets in ?(X, Y) when this space is endowed with the topology of uniform convergence on the class of all weakly null sequences. Finally, we show that similar arguments can be applied to the study of uniformly completely continuous sets. © 2007 Birkhäuser Verlag Basel/Switzerland.