By the duality theorems of Deligne, Doplicher and Roberts, two finite (or compact) groups are isomorphic if and only if their categories of representations are equivalent as symmetric monoidal categories. About ten years ago, it was discovered independently by several teams of authors that non-isomorphic finite groups can have equivalent monoidal categories of representations. In order to obtain a better understanding and allow generalization to infinite compact groups, we develop a somewhat more conceptual approach to the study of isocategorical groups.