We analyze two traces of TCP--traffic recorded at the gateway of a LAN corre­ sponding to two hours at Berkeley and to eight hours at CNET labs respectively. We are mainly interested in a multifractal approach, which we introduce with var­ ious examples. We elaborate on the difference to (mono)fractal statistical tests being used so far. Though we find statistical self­similarity, the data is clearly not monofractal. Consequently, previously proposed models based on fractional Brow­ nian motion are correct only up to second order statistics. Multifractal analysis allows several conclusions which could not be made with the usual estimation of the 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.