(Due to a technical defect the lecture could not be recorded completely.) The lecture was held within the framework of the Hausdorff Trimester Program: Applied and Computational Algebraic Topology. Andreas Thom: Algebraic vs. topological K-theory Abstract: I will start out giving a general introduction to K-theory and some basic applications to algebra, geometry and number theory. I will then discuss the comparison of algebraic and topological K-theory for specific rings of functional analytic type. In particular, I will discuss the algebraic K-theory of stable algebras and of rings of continuous functions. If time permits, I will prove a version of the homotopy invariance theorems for algebraic K-theory stabilised by a suitable operator ideal and for negative algebraic K-theory of rings of continuous functions.