The jurisdiction of international courts and tribunals and the admissibility of inter-State claims under international law are central to international adjudication, operating as a gateway to the litigation on the merits – the end goal of the proceedings. Still, these concepts remain inherently under-defined, and can be shaped in multiple ways to formulate preliminary objections in international litigation in general. International investor-State arbitration adds specific aspects and complexities to the issue. This introductory contribution accounts for the theoretical deficiencies underpinning the notions of jurisdiction and admissibility, with a special focus on international investment arbitration, and introduces the selected case-studies which form the subject-matter of the articles in this Special Issue. The recent Urbaser award is also used as an example of the unexplored potential of novel – and critical – legal argumentation relating to the jurisdiction of investment tribunals.