The Travelling Salesman Problem (TSP) represents one of the most extensively studied NP-hard combinatorial optimization problems in computer science and operations research [10,42]. This comprehensive survey examines the evolution, current state, and future directions of TSP algorithms and approaches, analyzing over 87 recent research contributions spanning exact algorithms, approximation methods, heuristics, metaheuristics, and emerging machine learning techniques [1,5]. Our analysis reveals signi cant advances in quantum computing approaches [1,4], machine learning integration [30,33], and hybrid optimization strategies [5,11]. Key ndings indicate that while Christo des' algorithm maintains its 1.5approximation ratio established in the 1970s [17,20], recent breakthroughs have achieved (1.5-ε) approximation for some constant ε > 10−36 [43]. The Lin-Kernighan heuristic and its variants remain the gold standard for practical TSP solving, with Concorde solver achieving optimal solutions for instances up to 85,900 cities [29,32]. Emerging quantum algorithms demonstrate exponential speedup potential with O(n3 log(n)) complexity [1], while machine learning approaches using graph neural networks show promising results for both constructive and improvement paradigms [30,33]. This survey provides a systematic analysis of algorithmic complexity