We define concepts of amenability and co-amenability for algebraic quantum groups in the sense of A. Van Daele [21]. We show that co-amenability of an algebraic quantum group always implies amenability of its dual. Various necessary and/or sufficient conditions for amenability or co-amenability are obtained. Co-amenability is shown to have interesting consequences for the modular theory in the case that the algebraic quantum group is of compact type. Subj. Class.: Ouantum groups, C*-algebras MSC 2000: Primary 46L05, 46L65. Secondary 16W30, 22D25, 58B32. Keywords: quantum group, amenability.