The object of this paper is to establish the fundamental equations and there approximation in nonlinear theory of thin elastic shells under the Kirchhoff-Love hypotheses as a basis for analysis of elastic stability and large deflection. The derived nonlinear equilibrium equations for general-shaped shells allowing for the effect of deformation on equilibrium are given in terms of the stresses and displacements. Substituting the constitutive equations into the above equilibrium equations, our fundamental equations are obtained by the strain measures and related quantities for shell. Each terms in rigorous and complex those equations are estimated by the introduction of the concept of estimations about the order of magnitude for the important parameters (i.e. the membrane strain, the bending strain and the wave length in the shell region). The discussion of consistent and systematic approximations corresponding to the basic assumption of small strains from the exact fundamental equations of nonlinear shell theory are given under the above estimation.