Abstract While function symbols are widely acknowledged as an important feature in logic programming, they make common inference tasks undecidable. To cope with this problem, recent research has focused on identifying classes of logic programs imposing restrictions on the use of function symbols, but guaranteeing decidability of common inference tasks. This has led to several criteria, called termination criteria, providing sufficient conditions for a program to have finitely many stable models, each of finite size. This paper introduces the new class of bounded programs which guarantees the aforementioned property and strictly includes the classes of programs determined by current termination criteria. Different results on the correctness, the expressiveness, and the complexity of the class of bounded programs are presented.