The Probe Of Extreme Multi-Messenger Astrophysics (POEMMA) is designed to accurately observe ultra-high-energy cosmic rays (UHECRs) and cosmic neutrinos from space with sensitivity over the full celestial sky. POEMMA will observe the air fluorescence produced by extensive air showers (EASs) from UHECRs and potentially UHE neutrinos above 20 EeV. Additionally, POEMMA has the ability to observe the Cherenkov signal from upward-moving EASs induced by Earth-interacting tau neutrinos above 20 PeV. The POEMMA spacecraft are designed to quickly re-orientate to follow up transient neutrino sources and obtain currently unparalleled neutrino flux sensitivity. Developed as a NASA Astrophysics Probe-class mission, POEMMA consists of two identical satellites flying in loose formation in 525 km altitude orbits. Each POEMMA instrument incorporates a wide field-of-view (45 ∘ ) Schmidt telescope with an optical collecting area of over 6 m 2 . The hybrid focal surface of each telescope includes a fast (1 μs) near-ultraviolet camera for EAS fluorescence observations and an ultrafast (10 ns) optical camera for Cherenkov EAS observations. In a 5-year mission, POEMMA will provide measurements that open new multi-messenger windows onto the most energetic events in the universe, enabling the study of new astrophysics and particle physics at these extreme energies.