We analyze two traces of TCP--traffic recorded at the gateway of a LAN corresponding to two hours at Berkeley and to eight hours at CNET labs respectively. We are mainly interested in a multifractal approach which we compare with the (mono)fractal statistical tests for self-similarity being used so far. Though we find evidence for statistical self-similarity in certain aspects of the traffic, the overall conclusion is that it is clearly not monofractal. Consequently, previously proposed models based on fractional Brownian motion are correct only up to second order statistics. Moreover, with multifractal measures simple simulations of traffic traces become available. Multifractal analysis, finally, allows several conclusions which could not be made with an estimation of the statistical self-similarity parameter $H$, called {\em Hurst exponent}. Among these we mention the remarkable fact that incoming and outgoing traffic observed at the same gateway may show strikingly different multifractal behavior. Furthermore, our analysis suggests that heavy and moderate outgoing traffic at Berkeley are in some sense independent of each other. Finally, the outgoing and the incoming traffic at Berkeley and at CNET look entirely different from a multifractal point of view.