The theory of planar involute gearing is generalized for the design and machine-cutting of spatial involute gearing. The monograph solves a problem, which has been discussed for over 150 years. Involute skew helicals, oblique involute hypoids, offset involute worms, bevels and straight spur gears are all explicable by the same unifying theory. A fundamental law, not only for involute but for all kinds of gearing, is enunciated and proven. The spatial involute gearing introduced and explained in this book involves in all cases straight line paths for both points of contact, constant angular velocity ratio, absence of any built-in transmission error, the non-importance of all minor errors at assembly, toleration of the flexing of an elastic gear box, and ample opportunity to avoid inadequate load bearing and lubrication. The continuous screw motion of an imaginary straight-sided rack generating contemporaneously both flanks of the spatial involute teeth can be readily related to ordinary engineering practice.