Regenerative cooling represents one of the most critical thermal management challenges in liquid-propellant rocket engine design, where combustion chamber temperatures exceeding 3000 °C necessitate sophisticated cooling strategies to prevent structural failure. This comprehensive paper examines the evolution of regenerative cooling systems from the early tubular designs of the Saturn V F-1 engine to contemporary monolithic structures fabricated through additive manufacturing techniques. We analyze the fundamental heat transfer mechanisms governing regenerative cooling, examine propellant selection criteria with emphasis on cryogenic oxidizers and hydrocarbon fuels, and evaluate the comparative advantages of conventional De Laval nozzle configurations versus aerospike designs. Special attention is devoted to emerging physics-based computational design methodologies that leverage first-principles understanding rather than machine learning approaches, exemplified by the Neiron algorithm developed by Leap71. Through rigorous thermodynamic analysis, materials science considerations, and examination of dual-propellant cooling strategies in aerospike engines, this work establishes the current state-of-the-art in regenerative cooling technology and identifies critical research directions for next-generation propulsion systems. The integration of computational design tools with metal additive manufacturing enables unprecedented geometric complexity and cooling efficiency, fundamentally transforming rocket engine development methodologies.