The mathematical theory of the scattering operator is developed for the general scattering systems involving an arbitrary number of channels. It includes as a special case the theory for 'simple scattering systems' given in an earlier paper. The scattering system is defined as a quantum mechanical system which satisfies certain asymptotic and completeness conditions given in Section 4. The existence of the S-operator as well as its unitary property is then a rigorous mathematical consequence of this property. A crucial step in these deductions is the orthogonality theorem for the left projections of the wave operators which is proved in Section 5. In the last Section 7, we discuss the various ways of introducing the 'in' and 'out'-operators and their relation to the S-operator.