On Amenability and Co-Amenability of Algebraic Quantum Groups and Their Corepresentations We introduce and study several notions of amenability for unitary corepresentations and $*$-representations of algebraic quantum groups, which may be used to characterize amenability and co-amenability for such quantum groups. As a background for this study, we investigate the associated tensor C$^{*}$-categories. Keywords:quantum group, amenabilityCategories:46L05, 46L65, 22D10, 22D25, 43A07, 43A65, 58B32