Logic programming under the stable model semantics has been extended to arbitrary formulas. A question of interest is how to characterize the property of well-supportedness, in the sense of Fages, which has been considered a cornerstone in answer set programming. In this paper, we address this issue by considering general logic programs, which consist of disjunctive rules with arbitrary propositional formulas in rule bodies. We define the justified stable semantics for these programs, propose a general notion of well-supportedness, and show the relationships between the two. We address the issue of computational complexity for various classes of general programs. Finally, we show that previously proposed well-supported semantics for aggregate programs and description logic programs are rooted in the justified stable semantics of general programs.