A sequence of 633 high spatial resolution magnetograms and continuum images from SOHO MDI of NOAA AR 0330 is used to study moving magnetic feature (MMF) activity in the moat surrounding a mature leader sunspot. The time-averaged frame shows that the moat region is covered by a magnetic field that exhibits the same polarity distribution as that observed in the penumbra. The moat field displays the true polarity of the spot in the sector where the penumbra displays it. Similarly, on the side where the penumbra shows a polarity opposite the true one (due to projection effects after the so-called apparent neutral line), the moat field also displays a polarity opposite the true one. This is only compatible with a moat field that is horizontal almost everywhere, as in the outer penumbra. Indeed, this horizontal moat field is seen to be physically connected with the penumbra. This connection is made evident when analyzing the individual structures detected in the averaged images, which we call moat filaments. The filaments stretch out for 12" in the moat and can be traced back into the penumbra. The observed polarity distribution along them is only compatible with mean inclinations in the range of 80°-90°. Inside the spot, these filaments are linked to the more horizontal magnetic field component that is thought to carry a large part of the Evershed flow. Several bipolar MMFs are seen to originate inside the penumbra and cross the sunspot outer boundary to enter the moat region, following the paths outlined by the moat filaments. These results are discussed in the frame of our current theoretical understanding of the Evershed flow and MMF activity.