At the Paul Scherrer Institute slow positive muons (μ⁺) with nearly 100% polarization and an energy of about 10 eV are generated by moderation of an intense secondary beam of surface muons in an appropriate condensed gas layer. These epithermal muons are used as a source of a tertiary beam of tunable energy between 10 eV and 20 keV. The range of these muons in solids is up to 100 nm which allows the extension of the μ⁺SR techniques (muon spin rotation, relaxation, resonance) to the study of thin films. A basic requirement for the proper interpretation of μ⁺SR results on thin films and multi-layers is the knowledge of the depth distribution of muons in matter. To date, no data are available concerning this topic. Therefore, we investigated the penetration depth of μ⁺ with energies between 8 keV and 16 keV in Cu/SiO₂ samples. The experimental data are in agreement with simulated predictions. Additionally, we present two examples of first applications of low energy μ⁺ in μ⁺SR investigations. We measured the magnetic field distribution inside a 500-nm thin High-T_C superconductor (YBa₂Cu₃O_7-δ), as well as the depth dependence of the field distribution near the surface. In another experiment a 500-nm thin sample of Fe-nanoclusters (diameter 2.4(4) nm), embedded in an Ag matrix with a volume concentration of 0.1%, was investigated with transverse field μ⁺SR.