Abstarct: In recent decades, fractional calculus has gained a prominent position in applied sciences and engi-neering as a generalization of classical calculus for modeling phenomena with memory and nonlocality. Despite the richness of the existing theory, the practical implementation of this concept has consistently required the development and analysis of accurate and efficient numerical methods due to the integral and complex nature of fractional operators. This thesis aims to provide a systematic review, comparative anal-ysis, and an integrated frxamework of numerical methods for computing fractional integrals and deriva-tives. In this thesis, existing numerical methods, such as interpolation-baxsed methods, spectral methods, fractional linear multistep methods and radial basis function-baxsed methods for computing fractional in-tegrals and derivatives, are categorized and analyzed and the foundations of error analysis, stability and convergence for each are examined. Then, focusing on the axiomatic foundations of fractional calculus, a precise definition of fractional derivatives within the frxamework of fractional integral-derivative oper-ators is established, and the fundamental theorem of calculus for these operators is proved. This thesis provides necessary clarity regarding the conditions for the validity of this theorem and the mathematical relationships between different definitions of fractional derivatives. By integrating method-oriented and principle-oriented perspectives, this research not only presents a list of numerical methods but also ex-plains the theoretical basis for selecting each method, its limitations and optimization criteria baxsed on the problem type, required accuracy and computational efficiency. Finally, this integrated frxamework can serve as a guide for researchers and engineers in selecting and implementing suitable numerical meth-ods for solving fractional differential equations in fields such as continuum mechanics, thermodynamics, signal processing and dynamical systems.