Abstract: Up to now, there is a need for a general theory of conjugate surfaces of continuous sliding contact. To try to satisfy this requirement is just tile aim of this paper. In the paper, a general and complete discussion of conjugate surfaces of continuous sliding contact, as starting from the fundamental conjugate conditions, is given. Owing to the adoption of pure analytical method instead of the customarily used grapho-analytical method (e. g. referring to literatures [1.2.3.4]), the analysis is applicable to broader extent. Restrictions, such as constant speed ratio, constant center distance, no axial motions, and parallelism or perpendicularity of two axes, are removed, and the analysis is adapted to more generally conjugate problems, as with variable speed ratio, with variable center distance, with axial motions, and with any angle between two axes. In consequence of more strictly logical nature of pure analytical method, many errors, produced in grapho-analytical method due to erroneous propositions, or to complexity of graphs, can be corrected. In this papers some errors in literaturesCl033 are discussed. Besides, some new conclusions, which till now, have not yet been rigorously proved, or not yet been reported, are investigated in this paper, such ass the constant-speed-ratio conjugate action of a pair of spiral involute gears (section 五，(三)), "elliptically conjugate" surfaces (section 六), equations expressing "property of interchangeability between revolution and displacement" (suction 七.) etc. Conjugate surfaces of continuous rolling contact, i.e. pitch surfaces, are treated as a degenerated case of continuous sliding contact; and the condition for their existence is formulated (section 八.(一)). At the end of this paper (section 九.), widespread applications of the conjugate theory in practical industries of production are enumerated. The generative motion for production of hypoid gears is formulated in a more straightforward and more rigorous manner, in comparison with literatures[5.6] (section 九(二)). For sake of simplification of mathematical formulae, vetor-methods are employed. The symbols for vector-calculation are all appropriated from the literature [7].The final results, however, are expressed in Cartesian coordinate system for facility in use of them.