The independence and impartiality of the judge or arbitrator are fundamental aspects of any adjudicative proceedings, whether at the domestic or international level. Such qualities may be affected by “particular situations” preventing a member of the court or tribunal from participating in a given case. This in turn gives rise to two types of incidental procedure: “recusal” (or disqualification) and “abstention,” both of which are domestic law mechanisms applied — not unreservedly — to international litigation. It should be added that the procedures in question do not have a single aim. Their purpose may be to ensure the impartiality of the proceeding (self-recusal or disqualification of the member of the adjudicative body) or to call into question the very possibility of conducting legal proceedings (spontaneous withdrawal of the arbitrator, or refusal to sit at the instigation of the appointing state). This article deals with these incidental procedures by examining the basic texts and the practice of international courts and tribunals.