The paper is concerned with the nonlinear dynamics of geometrically exact shells and spatial rods with an SO(3)-valued configuration space. Energy conserving/energy dissipating methods are to be discussed and dimension reduction and post-processing methods will be presented. An energy-momentum integration scheme is constructed which is designed so as to be independent of the configuration space, preserve, by design, specific features of the continuous system such as conservation of momentum, angular momentum, when the applied forces allow to, and conservation of total energy in case of a Hamiltonian system. The application of such methods is of importance since they allow for stable integration in long term dynamics and ensures that the dissipation is of a purely physical source, in case of dissipative systems. However, in certain cases high frequency dissipation is desirable and can be achieved by adding material viscosity-like terms to the dynamic equations. We discuss a dimension reduction method based on the Proper Orthogonal Decomposition Method. The method allows the deduction of the energetically dominating modes of the motion and reduces the motion to these essential modes. Further a post- processed Galerkin finite element method based on hierarchical interpolations is presented. The method makes use of two stress hybrid or, alternatively, enhanced strain finite element formulations; a low-dimensional four-node and a high-dimensional nine- node one. The results of the four-node element are projected on the nine-node ones using a specific projection operator. Altogether the method provides means for a very low-cost improvement of low-dimensional solutions. The kinematical fields are improved to higher orders which makes the method effective in handling large scale problems in structural dynamics.