This paper presents a stability analysis for the dynamic model of a travel route choice adjustment process proposed by Dupuis and Nagurney (1993). In particular, we investigate both local and global stability issues for this adjustment process that may be of concern to transportation planners and analysts. Two distinct approaches for stability analysis are proposed based on the general stability analysis results obtained recently by Zhang and Nagurney (1995) for projected dynamical systems. The global stability results are established using a “monotonicity” approach, under monotonicity assumptions on the link cost and travel disutility functions. A distinct approach, termed a “regularity” approach, is then provided to verify the local stability of the adjustment process at a regular equilibrium flow pattern