The basic concepts of a general method of kinematic synthesis of space mechanisms are developed by means of vectors and quaternion operators applicable to path, function, and motion generation (body guidance) for finite and infinitesimal displacements (point, order, and combined point-order approximations). For writing the position equations, space mechanisms are represented by one or more loops of a general kinematic chain of ball-jointed bar-slideball members. Appropriate mathematical constraints on the relative freedom of these members render the general chain equivalent to the represented mechanism. The method leads to a system of equations of canonical simplicity, uniform for all tasks of finite spatial synthesis, often yielding closed-form linear solutions for small numbers of precision conditions. The same system of equations is then used to refine the solution for greater precision by numerical methods. Typical applications are indicated, some involving the use of a spatial finite circlepoint-center point theory, which includes classical planar Burmester theory as one of its special cases. An earlier general complex-number method of planar synthesis is shown to be a special case of the general spatial method introduced here.