This work summarizes the results of my Ph.D. thesis on ordinal costs in combinatorial optimization. Ordinal costs model the quality of objects whenever numerical values are not appropriate or available. As an example the safety of a street for a cyclist can be ranked in ordered categories, like, e.g., safe (bike lane), medium safe (slow traffic) and unsafe (main road). Other examples for ordinal costs relate to quality, sustainability and rankings of, e.g., olympic medals. We consider ordinal costs for general combinatorial optimization problems and analyze the interrelation between ordinal optimization and multi-objective optimization. Moreover, in the thesis for the specific case of matroid optimization problems very efficient solution algorithms are suggested. In this paper the ideas of the solution methods and the main theoretical results are summarized.