This study presents a general higher-order shell theory for geometric and hyperelastic material nonlinearities. The main features of this shell theory are as follows: (i) the formulation utilizes an orthonormal moving frame (OMF) instead of the classical natural covariant frame, simplifying the mathematical formulation even for complex constitutive relations due to the orthonormal nature of the basis vectors; (ii) the general higher-order approximation through the thickness of the shell enables modeling of both thin and thick shell deformations, while also alleviating numerical locking associated with thickness stretch; and (iii) it includes constitutive relations for isotropic, transversely isotropic, anisotropic and morphoelastic shell structures. This theory is particularly useful for analyzing problems in vascular mechanics and other soft tissue structures.In this framework, the displacement field normal to the shell reference surface is approximated using general power series polynomials for single-layer shell structures. Subsequently, the shell kinematics for the orthonormal basis is derived. As shown by Arbind, Reddy, and Srinivasa in [1], such a coordinate system enables a more efficient representation of kinematic quantities (e.g., the determinant of the deformation gradient) compared to the classical tensorial representation with a covariant basis. This study extends the work in [1] to include more complex constitutive relation for the large deformation of soft shell structures. For general shell surface geometries, the weak form finite element formulation of the shell theory is presented, and the theory is further applied to large deformation analysis of curved structures, such as biological tubes under internal pressure. For incompressible materials, the incompressibility constraint is imposed using both the penalty method and the Lagrange multiplier method. Additionally, nonlinear finite element formulations for post-buckling analysis are presented, employing the arc-length method. Various numerical examples are provided to verify and validate the formulation presented in this study. Reference: [1] A Arbind et al. International Journal for Numerical Methods in Engineering, 122(1):235–269, 2021