We study the physics potential of the detection of the Cosmic Neutrino Back- ground via neutrino capture on tritium, taking the proposed PTOLEMY experiment as a case study. With the projected energy resolution of ∆ ∼ 0.15eV, the experiment will be sensitive to neutrino masses with degenerate spectrum, m₁ ≃ m₂ ≃ m₃ = m_ν 0.1eV. These neutrinos are non-relativistic today; detecting them would be a unique opportunity to probe this unexplored kinematical regime. The signature of neutrino capture is a peak in the electron spectrum that is displaced by 2m_ν above the beta decay endpoint. The signal would exceed the background from beta decay if the energy resolution is ∆ 0.7 mν. Interestingly, the total capture rate depends on the origin of the neutrino mass, being Γ^D ≃ 4 and Γ^M ≃ 8 events per year (for a 100 g tritium target) for unclustered Dirac and Majorana neutrinos, respectively. An enhancement of the rate of up to O(1) is expected due to gravitational clustering, with the unique potential to probe the local overdensity of neutrinos. Turning to more exotic neutrino physics, PTOLEMY could be sensitive to a lepton asymmetry, and reveal the eV-scale sterile neutrino that is favored by short baseline oscillation searches. The experiment would also be sensitive to a neutrino lifetime on the order of the age of the uni- verse and break the degeneracy between neutrino mass and lifetime which affects existing bounds.