The trace formula for an arbitrary connected reductive group over a number field was developed by James Arthur. The twisted case was the subject of the Friday Morning Seminar at the Institute for Advanced Study in Princeton during the 1983–1984 academic year. During this seminar, lectures were given by Laurent Clozel, Jean-Pierre Labesse and Robert Langlands. Having been written quite hastily, the lecture notes of this seminar were in need of being revisited. The authors' ambition is to give, following these notes, a complete proof of the twisted trace formula in its primitive version, i.e., its noninvariant form. This is a part of the project of the Parisian team led by Laurent Clozel and Jean-Loup Waldspurger. Their aim is to give a complete proof of the stable form of the twisted trace formula, and to provide the background for the forthcoming book by James Arthur on twisted endoscopy for the general linear group with application to symplectic and orthogonal groups.