The Payload for Ultrahigh Energy Observations (PUEO) long-duration balloon experiment is designed to have world-leading sensitivity to ultrahigh-energy neutrinos at energies above 1 EeV. Probing this energy region is essential for understanding the extreme-energy universe at all distance scales. PUEO leverages experience from and supersedes the successful Antarctic Impulsive Transient Antenna (ANITA) program, with an improved design that drastically improves sensitivity by more than an order of magnitude at energies below 30 EeV. PUEO will either make the first significant detection of or set the best limits on ultrahigh-energy neutrino fluxes.