This is the first half of a two-part essay on jurisdiction and admissibility in investment arbitration. It focuses on the arbitration practice, whilst the second part sets these concepts in the wider framework of public international law litigation. This essay maps the objections to the tribunal’s jurisdiction (by ratio: materiae, temporis, loci and personae) and the claim’s admissibility. It offers some preliminary conclusions: in certain areas there still is no consensus; tribunals are inclined to characterise objections as jurisdictional, and rarely resort to admissibility; findings of inadmissibility draw a judgment on the claimant or the claim’s propriety (whilst jurisdictional decisions typically eschew value-judgment); tribunals failed to distinguish jurisdiction from admissibility. These findings are further explored, within a wider theoretical context, in the second part of the essay.