GRB 030329 is unique in many aspects. It has a very low redshift for a GRB, z = 0.1685, and is therefore very bright and easy to monitor, making it the most well studied afterglow to date. It shows a supernova bump in the light curve, with a spectrum very similar to SN 1998bw, thus establishing with much better confidence the connection between GRBs and core collapse SNe. There are also two important physical characteristics that make this burst especially interesting, aside from its remarkably low redshift. First, unlike most GRB afterglows, the light curve of GRB 030329 shows a very large variability a few days after the burst. These fluctuations show a roughly constact amplitude, and a constant duration t, while t/t decreases with time t. Second, its -ray energy output and X-ray luminosity at 10 hr are a factor of 20 and 30, respectively, below the average value around which most GRBs are narrowly clustered. We consider several interpretations for the variability in the light curve, in the context of different physical mechanisms, and find that the most likely cause is refreshed shocks, i.e. slow shells that are ejected from the source and catch up with the afterglow shock at late times. In GRB 030329 this happens after the jet break, which implies an approximately constant duration t of the bumps, in agreement with the observations. This interpretation also explains the anomalously low initial energy of this burst, as the total energy of the afterglow shock is increased by a factor of 10 due to the refreshed shocks, thus bringing the total energy output close to the average value for all GRBs.