Traffic assignment is a key component in transport planning models. It models travel behaviour in terms of route choice. This is essential to accurately forecast travel demand and most importantly to enable the correct assessment of the benefits of changes in transport policies and infrastructure developments. The route choice of travellers may be influenced by multiple objectives, for example travel time but also travel associated toll costs. Here, travellers may avoid a fast route because of toll costs associated with it. We explicitly distinguish those functions as separate route choice objectives. This leads to the concept of multi-objective traffic assignment (MTA). We discuss the concept of MTA, and develop heuristic solution methods to obtain equilibrium solutions of MTA and present some computational results.