Using the concept of the p-capacity, associated with uniform Sobolev spaces L1 p (Ω), we introduce a notion of a p-capacitory boundary for an arbitrary 1 domain Ω ⊂ R n, n − 1 < p ≤ n that represent ”ideal boundaries” of domains Ω ⊂ Rn for p-capacitory metrics. The Sobolev classes L 1 p (Ω) can be extended to the p-capacitory boundaries under some natural assumptions on Ω. The p-capacitory topology is equivalent to the Euclidean one into Ω, but the p-capacitory boundary depends on p and can be very far from the Euclidean one for non-regular domains. An analog of p-capacitory boundaries can be introduced for p > n also using a more delicate procedure. Joint work with Alexander Ukhlov.