The talk will give an overview on recent developments on the asymmetric traveling salesman problem (ATSP). In contrast to the symmetric problem variant, where the Christofides-Serdyukov algorithm gives a simple 3/2-approximation, no constant-factor approximation algorithm has been known for a long time. The talk will give an overview of the different approaches that have lead to improved approximation guarantees over the last decade. I will present the approach of our paper with Svensson and Tarnawski that led to the first constant factor approximation guarantee in 2017, and a subsequent improvement by Traub and Vygen.