Algebraic $K$-theory, assembly maps, controlled algebra, and trace methods

Space - Time - Matter. Analytic and Geometric Structures, pp. 1-50, De Gruyter, 2018 We give a concise introduction to the Farrell-Jones Conjecture in algebraic $K$-theory and to some of its applications. We survey the current status of the conjecture, and we illustrate the two main tools that are used to attack it: controlled algebra and trace methods.