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The paper introduces the principle Maximise Presupposition and its cognates. The main focus of the literature and this article is on the inferences that arise as a result of reasoning with Maximise Presupposition ('anti-presuppositions'). I will review the arguments put forward for distinguishing them from other inference types, most notably presuppositions and conversational implicatures. I will zoom in on three main issues regarding Maximise Presupposition and these inferences critically discussed in the literature: epistemic strength(ening), projection, and the role of alternatives. I will discuss more recent views which argue for either a uniform treatment of anti-presuppositions and implicatures and/or a revision of the original principle in light of new data and developments in pragmatics.