Abstarct: Abstract This thesis presents numerical approaches for solving variable-order fractional differential equations, which can model complex phenomena with memory dependent on time, space, or concentration. First, a comprehensive review of the mathematical foundations, physical models, and existing numerical methods for these equations is provided. Then, a new explicit finite difference method is developed for solving linear and semi-linear initial-boundary value problems baxsed on the variable-order Caputo derivative, and its conditional stability is mathematically proven. Subsequently, for solving a more practical class of problems, the Adams-Bashforth-Moulton algorithm is extended to solve variable-order fractional delay differential equations and the associated error analysis is presented. The efficiency and accuracy of both proposed numerical methods are validated by solving diverse examples from the fields of anomalous diffusion, population dynamics, and chemical kinetics, both numerically and graphically, and their capability to represent complex behaviors such as chaos is investigated