Let \(A\) be a Banach algebra and \(B\) a Banach \(A\)-bimodule. The purpose of the paper under review is the study of the \(n\)th dual \(A^{(n)}\) (resp. \(B^{(n)}\)) of \(A\) (resp. \(B\)), where \(n\) is an even nonnegative integer (\(B^{(n)}\) is a Banach \(A^{(n)}\)-bimodule). The main results established are characterizations of the topological centers of the module actions of \(A^{(n)}\) on \(B^{(n)}\), which extend previous results of A. T.-M. Lau and A. Ülger [Trans. Am. Math. Soc. 348, No. 3, 1191–1212 (1996; Zbl 0859.43001)]. The authors also prove that, under certain conditions involving weak topologies, such topological centers coincide with \(A^{(n)}\) or \(B^{(n)}\). The prerequisites for understanding the paper may be found in the text.