Abstract. Let K be a self-similar set in R d , of Hausdorff dimension D, and denote by |K( )| the d-dimensional Lebesgue measure of its -neighborhood. We study the limiting behavior of the quantity −(d−D) |K( )| as → 0. It turns out that this quantity does not have a limit in many interesting cases, including the usual ternary Cantor set and the Sierpinski carpet. We also study the above asymptotics for stochastically self-similar sets. The latter results then apply to zero-sets of stable bridges, which are stochastically self-similar (in the sense of the present paper), and then, more generally, to level-sets of stable processes. Specifically, it follows that, if Kt is the zero-set of a realvalued stable process of index α ∈ (1, 2], run up to time t, then −1/α |Kt( )| converges to a constant multiple of the local time at 0, simultaneously for all t ≥ 0, on a set of probability one.The asymptotics for deterministic sets are obtained via the renewal theorem. The renewal theorem also yields asymptotics for the mean E[|K( )|] in the random case, while the almost sure asymptotics in this case are obtained via an analogue of the renewal theorem for branching random walks.