Given an instance, "Search Problems" require finding a solution or proving that no solutions exist. Finding the shortest cycle in the graph, finding the occurrence of the given pattern in the given dataset are examples of the searching problem. The travelling salesman's task is to determine the optimal travel route for the salesman, whose goal is to visit all the cities prescribed in the task, in the shortest possible time, and at the lowest cost. In graph-theoretical terms, given a classical weighted complete graph G, find a minimum-weight Hamilton cycle of G. The TSP is classified as a NP-hard optimisation problem, that in the best case we may expect of polynomial-time algorithm which always returns a solution close to the optimal one. In section 2 of this work, the origins and history of the TSP are discussed, along with applications to indicate its significance. Since it is a problem in graph theory, corresponding terminology and notations are introduced in section 3. Later in section 4, the Hamiltonian cycle and conditions for its existence in a graph are described. Complexity and the terms P and NP are defined in section 5. In section 6, different approaches and algorithms for the solution of this problem are reviewed and compared.