The PSI Positron Production experiment, known as P3 or P-cubed, is a proof-of-principle positron source and capture system that can greatly improve the state-of-the-art positron yield. The P3 project is led by the Paul Scherrer Institute (PSI) under the auspices of CHART, a Swiss collaboration for accelerator research and technology. The experiment addresses the long-standing challenge faced by conventional injector facilities to generate, capture, and damp the emittance of high-current positron beams, which is a major limiting factor for the feasibility of future electron-positron colliders. P3 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 that has the potential to outperform significantly the present positron capture efficiency rates. The P3 experiment will be hosted at PSI's SwissFEL, and will serve as the positron source test facility of CERN's FCC-ee. It will use SwissFEL's 6 GeV electron drive beam and a tungsten target to make up a pair-production-driven positron source. Instead of a normal conducting flux concentrator, P3 will use a high-temperature superconducting (HTS) solenoid that will surround the target with a peak field above 12 Tesla, the highest ever used in a positron capture system. In addition, the large aperture of the HTS solenoid design allows for a full immersion of the target in the magnetic field, enabling the optimal conditions for positron capture. The target will be followed by a small linac consisting of two RF accelerating structures based on a novel standing wave solution with a large iris aperture, surrounded by sixteen normal conducting solenoids, which will deliver a nearly flat magnetic channel around 0.45 Tesla. The experiment diagnostics will consist of two arrangements of broadband pick-ups, two Faraday cups and various scintillator screens and fibers. Each setup will look at different characteristics of the secondary positron and electron distributions, including the charge, time structure, and energy spectrum. The diagnostics will help researchers optimize the operation parameters and will ultimately demonstrate the foreseen positron yield enhancement. This thesis provides a detailed account of the P3 experiment. It presents the concept design of P3, extending it to the implementation of the experiment's key accelerator technology and infrastructure. In addition, it reviews a comprehensive optimization study of the positron source and capture system, based on simulations of particle-matter interactions and beam dynamics. It also explains in detail the development of the above-mentioned diagnostics setups. Finally, two side activities are discussed: the beam tests of the broadband pick-ups at the CLEAR facility at CERN, and the development of a new concept for pair-production-driven positron production scheme based on conical converter targets. The calculations anticipate that, if the experimental findings live up to the expectations, P3 will demonstrate a four-fold increase in SuperKEKB's positron yield normalized to the drive linac energy, considered the current state-of-the-art. This implies that, in addition to other innovative features, P3 would make a substantial contribution to the accelerator and high-energy physics community by strengthening the feasibility of FCC-ee and other future colliders.