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We discuss a novel approach to measure the electron phase-relaxation length and femtosecond lifetimes at surfaces. It relies on the study of the spatial decay of quantum interference patterns in the local density of states (LDOS) with the STM. The method has been applied to s-p derived surface-state electrons on Cu(111) and Ag(111). The characteristic decay length of the LDOS oscillations is influenced by the finite lifetime, and thus reveals information about inelastic scattering in the two-dimensional (2D) electron gas. After an introduction in Section 1, we present a model describing the decay of Friedel oscillations off from straight steps in Section 2. Energy dependent lifetime measurements of hot electrons are presented in Section 3 and interpreted in terms of electron-electron scattering. The temperature dependent lifetime measurements of low-energy quasiparticles discussed in Section 4 give insight into the interaction of these 2D electrons with phonons. Our results on inelastic lifetimes are discussed in comparison with high-resolution angle-resolved photoemission and femtosecond two-photon photoemission measurements. (C) 2000 Elsevier Science B.V. All rights reserved.