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A problem that arises from time to time in international arbitration proceedings is how an arbitral tribunal is to deal with the evidence of a non-appearing witness. The applicable rules of arbitration typically confirm that the tribunal has the authority to determine (e.g.) the “admissibility, relevance, materiality and weight of the evidence offered,” but they rarely go any further than this. The purpose of this article is to seek to provide guidance to tribunals by reviewing the available arbitral practice. This reveals that there are circumstances in which tribunals have adopted broadly consistent approaches to the status of the non-appearing witness's written testimony (although this does not appear to extend to the treatment of a non-appearing expert's evidence), and that tribunals have generally not distinguished between the existence of a valid reason for a witness's failure to appear, or of exceptional circumstances which would warrant the witness's evidence being admitted. In addition, tribunals appear to be reluctant to place much weight on a non-appearing witness's written evidence unless it is corroborated by the testimony of other witnesses or documentary evidence, and they are also reluctant to draw adverse inferences from a witness's non-appearance.