We consider positive rules in which the conclusion may contain existentially quantified variables, which makes reasoning tasks (such as Deduction) undecidable. These rules, called forall-exist-rules, have the same logical form as TGD (tuple generating dependencies) in databases and as conceptual graph rules. The aim of this paper is to provide a clearer picture of the frontier between decidability and non-decidability of reasoning with these rules. We show that Deduction remains undecidable with a single forall-exist--rule; then we show that none of the known abstract decidable classes is recognizable. Turning our attention to concrete decidable classes, we provide new classes and classify all known classes by inclusion. Finally, we study, in a systematic way, the question "given two decidable sets of forall-exist--rules, is their union decidable?", andprovide an answer for all known decidable cases except one.