The term ‘ambition’ appears to have infiltrated international legal discourses: it is used to, for instance, lament the lack of state action to tackle major global challenges, praise progress towards difficult goals, or evaluate the outcomes of international law-making processes. Often mobilized, the concept of ambition in international law remains, however, poorly understood. And yet, each narrative offers a specific analytical frame that influences our understanding of the world and sets distinct policy prescriptions. What argumentative functions do ambition narratives play and what implications do they carry for international law, in both its practice and study? To respond to this question, the article explores the occurrence of the term in a field where the rationale of ambition has recently taken centre stage – international climate law – and uses the crisis narrative as a means of comparison to highlight the specificity of ambition discourses. The argumentative implications of ambition are identified in terms of vision, means and temporality: this article suggests that an ambition discourse fulfils objectives that a crisis narrative is unable to accommodate by calling for structural transformations, motivating states to commit to far-reaching objectives and adopting a long-term perspective focused on incremental change. The shortcomings of an ambition narrative are also highlighted, in relation to its determination and evaluation. The study contributes to shedding light on a new international law discourse to offer a different analytical frame for the discipline.