The Probe Of Extreme Multi-Messenger Astrophysics (POEMMA) is designed to identify the sources of Ultra-High-Energy Cosmic Rays (UHECRs) and to observe cosmic neutrinos, both with full-sky coverage. Developed as a NASA Astrophysics Probe-class mission, POEMMA consists of two spacecraft flying in a loose formation at 525 km altitude, 28.5 deg inclination orbits. Each spacecraft hosts a Schmidt telescope with a large collecting area and wide field of view. A novel focal plane is optimized to observe both the UV fluorescence signal from extensive air showers (EASs) and the beamed optical Cherenkov signals from EASs. In POEMMA-stereo fluorescence mode, POEMMA will measure the spectrum, composition, and full-sky distribution of the UHECRs above 20 EeV with high statistics along with remarkable sensitivity to UHE neutrinos. The spacecraft are designed to quickly re-orient to a POEMMA-limb mode to observe neutrino emission from Target-of-Opportunity (ToO) transient astrophysical sources viewed just below the Earths limb. In this mode, POEMMA will have unique sensitivity to cosmic neutrino tau events above 20 PeV by measuring the upward-moving EASs induced by the decay of the emerging tau leptons following the interactions of neutrino tau inside the Earth.