The agreement of disputing parties to each make a unilateral appointment of an arbitrator is among the most distinctive features of arbitral practice. A detailed examination, long overdue, of how this feature affects the actual process of arbitration is presented in this book. The study includes a historical analysis of unilateral nominations, a critical assessment of how the unilateral appointments system currently works and an empirical study of challenges of arbitrators. The author's critical assessment addresses several issues including: - limits to the right of the parties to make unilateral appointments; - the principle of equality of the parties in the constitution of the arbitral tribunal; - arbitrators' duty to be impartial and independent; - specific problems of bias in tribunals with party-appointed members; - the question of whether a different standard of impartiality and independence in party-appointed arbitrators makes any sense; - the presumption that party-appointed arbitrators can do things that presiding arbitrators cannot; and - the question of whether it is worth keeping the system of unilateral appointments as the default method for the constitution of multiple-member tribunals, or keeping it at all. The empirical study, in which the author offers a comparative analysis of challenges of arbitrators taking into account the method of appointment of the arbitrator, reveals interesting differences and coincidences between party-appointed and non-party-appointed arbitrators. The book ends with some suggestions on how the system of unilateral appointments could be improved, namely in order to increase the trust of each party in the arbitrator appointed by the other party and to allow an accurate match between what arbitration end-users may want from party-appointed arbitrators and what they ultimately get. For both its thorough and well-informed analysis and its sound recommendations, the book is sure to be welcomed by professionals in the arbitral community worldwide, as well as by arbitration law academics.