In higher-dimensional algebra, it is clear how to define \(n\)-categories (in the strict sense), but it is not clear how to define weak \(n\)-categories (which may well be more important). As a consequence, it is not clear how to define semistrict monoidal 2-categories or braided monoidal 2-categories. M. M. Kapranov and V. A. Voevodsky [J. Pure Appl. Algebra 92, No. 3, 241–267 (1994; Zbl 0791.18010)] have given very long definitions; the authors of this paper give definitions which are not quite equivalent but are shorter and more conceptual. Using their definitions, they show how to construct semistrict braided monoidal 2-categories as the centres of semistrict monoidal categories, and they give a strictification theorem for braided monoidal 2-categories. They also discuss applications to 4-dimensional topological quantum field theories and 2-tangles.