This chapter discusses – as a matter of comparative law, custom, arbitration rules, soft law and generally adopted practices – the roles and powers attributed to the chairperson and to the party-appointed arbitrators in tribunals composed of three members. The author first examines the evolution in the applicable standards of independence and impartiality with regard to presiding and party-appointed arbitrators; he then considers the express and implied powers bestowed upon, respectively, presiding and party-appointed arbitrators in the course of the proceedings, starting with the tribunal constitution process, addressing also the limits of permitted delegation of powers within tribunals, as well as the distinguishing characteristics of the function of the umpire in the legal systems that provide for it. In the last part of the chapter, the author focuses on the controversy as to the desirability of party-appointments in the contemporary law and practice of international arbitration.