In common shape optimization routines, deformations of the computational mesh usually suffer from decrease of mesh quality or even destruction of the mesh. To mitigate this, we propose a theoretical framework using so-called pre-shape spaces. This gives an opportunity for a unified theory of shape optimization, and of problems related to parameterization and mesh quality. With this, we stay in the free-form approach of shape optimization, in contrast to parameterized approaches that limit possible shapes. The concept of pre-shape derivatives is defined, and according structure and calculus theorems are derived, which generalize classical shape optimization and its calculus. Tangential and normal directions are featured in pre-shape derivatives, in contrast to classical shape derivatives featuring only normal directions on shapes. Techniques from classical shape optimization and calculus are shown to carry over to this framework, and are collected in generality for future reference. A pre-shape parameterization tracking problem class for mesh quality is in- troduced, which is solvable by use of pre-shape derivatives. This class allows for non-uniform user prescribed adaptations of the shape and hold-all domain meshes. It acts as a regularizer for classical shape objectives. Existence of regularized solu- tions is guaranteed, and corresponding optimal pre-shapes are shown to correspond to optimal shapes of the original problem, which additionally achieve the user pre- scribed parameterization. We present shape gradient system modifications, which allow simultaneous nu- merical shape optimization with mesh quality improvement. Further, consistency of modified pre-shape gradient systems is established. The computational burden of our approach is limited, since additional solution of possibly larger (non-)linear systems for regularized shape gradients is not necessary. We implement and com- pare these pre-shape gradient regularization approaches for a 2D problem, which is prone to mesh degeneration. As our approach does not depend on the choice of forms to represent shape gradients, we employ and compare weak linear elasticity and weak quasilinear p-Laplacian pre-shape gradient representations. We also introduce a Quasi-Newton-ADM inspired algorithm for mesh quality, which guarantees sufficient adaption of meshes to user specification during the rou- tines. It is applicable in addition to simultaneous mesh regularization techniques. Unrelated to mesh regularization techniques, we consider shape optimization problems constrained by elliptic variational inequalities of the first kind, so-called obstacle-type problems. In general, standard necessary optimality conditions cannot be formulated in a straightforward manner for such semi-smooth shape optimization problems. Under appropriate assumptions, we prove existence and convergence of adjoints for smooth regularizations of the VI-constraint. Moreover, we derive shape derivatives for the regularized problem and prove convergence to a limit object. Based on this analysis, an efficient optimization algorithm is devised and tested numerically. All previous pre-shape regularization techniques are applied to a variational inequality constrained shape optimization problem, where we also create customized targets for increased mesh adaptation of changing embedded shapes and active set boundaries of the constraining variational inequality.