Time-dependent traffic equilibrium problem gorverned by quasimonotone cost functions are studied. We prove, under weak regularity assumptions, the existence of Wardrop equilibrium flows. Those results extend previous ones of [P. Daniele, A. Maugeri, and W. Oettli, Time-dependent traffic equilibria, J. Optim. Theory Appl. 103(3) (1999), pp. 543-555] and [A. Nagurney, D. Parkes, and P. Daniele, The internet, evolutionary variational inequalities, and the time-dependent braess paradox, Comput. Manag. Sci. 4 (2007), pp. 355-375] in which pseudomonotonicity of the cost function was assumed. They are based on reformulation of the traffic network problem in terms of variational inequalities or quasivariational inequalities. Nonelastic and elastic cases are considered. © 2012 Copyright Taylor and Francis Group, LLC.