We consider the concept of triangulation of an oriented matroid. We provide a definition which generalizes the previous ones by BilleraMunson and by Anderson and which specializes to the usual notion of triangulation (or simplicial fan) in the realizable case. Then we study the relation existing between triangulations of an oriented matroid \(\mathcal{M}\) and extensions of its dual \(\mathcal{M}^*\), via the socalled lifting triangulations. We show that this duality behaves particularly well in the class of Lawrence matroid polytopes. In particular, that the extension space conjecture for realizable oriented matroids is equivalent to the restriction to Lawrence polytopes of the Generalized Baues problem for subdivisions of polytopes. We finish by showing examples and a characterization of lifting triangulations. Readership Graduate students and research mathematicians interested in convex and discrete geometry. Table of Contents  Introduction
 Preliminaries on oriented matroids
 Triangulations of oriented matroids
 Duality between triangulations and extensions
 Subdivisions of Lawrence polytopes
 Lifting triangulations
 Bibliography
