Explore cutting-edge research on the Traveling Salesman Problem (TSP) in this 40-minute conference talk from STOC 2020. Delve into the Asymmetric Traveling Salesman Problem, examining theorems, algorithms, and state-of-the-art approaches. Investigate lower bounds, past TSP reductions, and approximation results. Consider the challenges of Path TSP and its potential increased difficulty. Gain insights into key technical contributions and engage with thought-provoking questions. Conclude with an exploration of Half Integral TSP, broadening your understanding of this fundamental problem in computer science and optimization.