It is well-known that any finite triangulation possesses a unique maximal Schnyder wood. We introduce Schnyder woods of infinite triangulations, and prove there exists a unique maximal Schnyder wood of any infinite triangulation with finite boundary, and of the uniform infinite half-planar triangulation. Furthermore, the maximal Schnyder wood of the uniform infinite planar triangulation is the limit of maximal Schnyder woods of large finite random triangulations. Several structural properties of...