The elastostatic problem of two materially dissimilar wedges of arbitrary angles that are bonded together along a common edge and subjected to surface tractions on the boundary of the configuration is treated. Emphasis is placed on the investigation of the dependence of the order of the singularity in the stress field at the apex on the wedge angles and material constants. Numerical results for several special wedge geometries are included for all physically relevant material combinations. In particular the families of (a) equal angle wedges, (b) composite half planes, (c) composite full-planes, as well as the single geometry of a (d) half-plane bonded to a quarter-plane, are studied in detail.