Self-attracting walks (SATW) with attractive interaction u > 0 display a swelling-collapse transition at a critical u_c for dimensions d >= 2, analogous to the Θ transition of polymers. We are interested in the structure of the clusters generated by SATW below u_c (swollen walk), above u_c (collapsed walk), and at u_c, which can be characterized by the fractal dimensions of the clusters d_f and their interface d_I. Using scaling arguments and Monte Carlo simulations, we find that for uu_c, the

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... lusters are compact, i.e. d_f=d and d_I=d-1. At u_c, the SATW is in a new universality class. The clusters are compact in both d=2 and d=3, but their interface is fractal: d_I=1.50±0.01 and 2.73±0.03 in d=2 and d=3, respectively. In d=1, where the walk is collapsed for all u and no swelling-collapse transition exists, we derive analytical expressions for the average number of visited sites and the mean time to visit S sites.