Auswertung der Approximationsgüte durch positive lineare Operatoren Abstract: The areas of research covered by this thesis can be roughly divided into three parts: aspects of quantitative approximation, studying of shape-preservation properties and over-iteration for some selected operators. Regarding the quantitative approximation we study traditional problems like: direct estimates, degree of simultaneous approximation or global smoothness preservation (Chapters 2, 3). On the other hand, we also present some non-classical issues like: estimates for the Peano remainder, a quantitative Voronovskaja theorem or estimates for differences of two positive linear operators (Chapter 5). Our object of study are different classes of operators: rational type operators, composite Beta type operators, but also other types that cannot be classified like: the BLaC-wavelet operator and the King operator.