Summary: We study the model of traffic flow proposed by {\it A. Aw} and {\it M. Rascle} in [SIAM J. Appl. Math. 60, No.~3, 916‒938 (2000; Zbl 0957.35086)]. We constructed weak global in time solution using the Glimm method, we also constructed weak global in time solution using the wave-front tracking method. In both cases, we assume that the total variation of the initial data is locally bounded and we do not consider the appearance of the vacuum.