The PSI Positron Production experiment, known as P\textsuperscript{3} or \textit{P-cubed}, is a proof-of-principle positron source and capture system that can greatly improve the state-of-the-art positron yield. The P\textsuperscript{3} project is led by the Paul Scherrer Institute in Switzerland, and addresses the long-standing challenge faced by conventional injector facilities to generate, capture, and damp the emittance of high-current positron beam, which is a major limiting factor for the feasibility of future electron-positron colliders. P\textsuperscript{3} follows the same basic principles as its predecessors, utilizing a positron source driven by pair-production and an RF linac with a high-field solenoid focusing system. However, it incorporates pioneering technology, such as high-temperature superconducting solenoids, that can outperform significantly the present positron capture efficiency rates. The P\textsuperscript{3} experiment will be hosted at PSI's SwissFEL, and will serve as the positron source test facility of CERN's FCC-ee. This paper outlines the concept, technology, infrastructure, physics studies and diagnostics of P\textsuperscript{3}.