High-order Runge–Kutta (RK) methods are widely used for time integration in the numerical solution of time-dependent partial differential equations (PDEs), offering a balance between computational efficiency and accuracy. This article presents a comprehensive review of error analysis and stability properties of high-order RK schemes in the context of time-dependent PDEs. We discuss the method of lines framework, local and global error behavior, and the interplay between spatial and temporal discretization. Special attention is given to stability analysis, including strong stability preserving (SSP) properties, step-size restrictions, and the performance of explicit, implicit, and IMEX RK schemes for stiff and non-stiff problems. Numerical results and theoretical insights are provided to illustrate the strengths and limitations of these methods.