In this chapter, we develop a theory of shells that we believe is simple yet comprehensive and free of unnecessary assumptions. Thus, as with the special case of beamshells and axishells discussed in Chapters IV and V, we first derive exact equations of motion merely by specializing to shell-like bodies the integral-impulse form of the equations of motion of a three-dimensional continuum. This reduction, which involves no kinematic hypotheses (such as those associated with the names of Kirchhoff or Cosserat) and introduces no expansions through the thickness, leads to two-dimensional stress results and couples, a deformed position vector, and a rotational momentum vector, all expressed as certain weighted integrals with respect to a thicknesslike coordinate. In doing so, we introduce neither a midsurface nor so-called shell coordinates. Exact, conjugate strain measures fall out naturally and automatically from a mechanical work identity. As we emphasize throughout this book, the approximate nature of nonlinear shell theory enters only through the constitutive relations which—as in three-dimensional elasticity—ultimately rest on experimental data. It is at this point (and not before) that we introduce the famous Kirchhoff Hypothesis, now free of the contradictions it engenders if stated as a strict kinematic hypothesis. After a thorough discussion of the field equations and boundary conditions implied by this hypothesis, we end with a classification and discussion of various additional approximations introduced into shell theory to simplify the governing equations, including those leading to quasi-shallow shell theory and (shell-) membrane theory, the subject of the previous chapter.