Partial differential equations (PDEs) are essential mathematical tools used to model a wide range of physical processes in science and engineering, such as heat flow, fluid motion, electromagnetism, and quantum mechanics. Yet their efficient solution remains a significant computational challenge, especially for high-dimensional or complex systems. Recent advances in quantum computing have spurred the development of quantum algorithms tailored to PDEs, with particular focus on quantum linear system solvers (QLSAs) and variational quantum algorithms (VQAs) such as the Quantum Approximate Optimization Algorithm (QAOA). We provide a comprehensive review of quantum approaches for solving PDEs, analyzing both QLSA-based and variational methods, and examining their theoretical and practical performance relative to classical techniques. We highlight the theoretical promise of QLSAs, including their potential for improved scaling in specific metrics, while acknowledging recent analysis showing that practical quantum advantage over classical methods remains challenging to achieve. We also examine the suitability of VQAs for current noisy intermediate-scale quantum (NISQ) devices. Furthermore, we discuss hybrid frameworks that integrate quantum algorithms with physics-informed neural networks (PINNs), demonstrating improved convergence and accuracy in benchmark problems. Our analysis underscores the promise of quantum computing for PDEs, particularly in domains closely aligned with quantum physics, while also identifying key challenges and future directions for achieving quantum advantage in practical applications.