Additive manufacturing is a very complex process. Hence, optimization models and methods play a central role for improving related technologies to save resources like material, time, and energy, and to protect the environment. We first provide existing principles for modeling corresponding tasks as problems of packing optimization, particularly the phi-function technique as a basic analytical tool. Due to the large number of items to pack, the mutual influence of items, and the resulting nonconvexity of the packing optimization problem, a practical solution may become extremely difficult. Therefore, in a generalized way, we present model-based algorithmic ideas to tackle such optimization problems. This includes a decomposition principle and heuristics for obtaining good feasible solutions. We discuss three main cases of packing optimization problems with significant relevance to additive manufacturing, where the general methodology is applied. In the first case we are concerned with the generation of void structures in components that should be printed. Those structures can save material, time, and energy. The second case deals with optimal placements of components in the build space of a 3D printer. In this way, production time can be saved. Finally, we study the problem of simulating the filling of a given 3D volume with particles of possibly different shapes. This may be employed for decreasing the porosity in printed parts, as well as for saving material.