Development and evaluation of freeform surface descriptions In recent years great efforts can be observed in the community of optical design to use the additional degrees of freedom afforded by freeform surfaces in optical systems. In particular one important question at the beginning of the concept and design of a freeform system is the decision how to mathematically describe the surface itself. If the development of an optical system with freeform surfaces is considered not only the optical design but also the mechanical design the manufacturing and the assembly of the component inside the whole system are important. The focus in this PhD is on the design phase. Therefore only functions are considered which are globally defined on the computational area inside the boundary. The main approaches of describing freeform surfaces are discussed from a more mathematical point of view. More two new polynomial sets the A polynomials 1st and 2nd kind are introduced. The choice of description is often limited by the available options in the commercial software. Therefore a systematic study of the different options can still not be found. To overcome this problem the selected descriptions are evaluated in a comprehensive assessment. The results are collected in the form of a benchmark which compares different freeform surface descriptions for optical systems with one freeform for various applications with different types of symmetry including refractive reflective and catadioptric systems. The representations under the viewpoint of different initial systems symmetry of the system sensitivity in optimization and quality of the final result are compared and discussed. In particular the influence of the basic shape selection the basic geometry to be Cartesian or polar and the importance of the orthogonality are investigated. Finally in a conclusion the major results are summarized and a recommendation for practical work is formulated.