Abstarct: In this thesis, we systematically study and develop spectral numerical methods baxsed on Jacobi polynomials and their generalizations. First, we introduce a broad family of generalized Jacobi polynomials and functions with arbitrary real indices. These functions are orthogonal under the corresponding Jacobi weight and retain the most important useful properties of classical Jacobi polynomials for spectral approximation. These functions are used as basis functions in the spectral Galerkin numerical method to solve ordinary differential equations and partial differential equations with homogeneous boundary conditions. This application offers three main advantages: precise mathematical investigation and analysis, optimal and fast implementation, and the solution of a system of matrix equations with favorable numerical conditioning, ultimately leading to better error estimates for the approximate solution. Next, we address the solution of fractional Volterra integro-differential equations. Using the Jacobi spectral collocation method and transforming the problem into an integral equation, an efficient numerical algorithm baxsed on Jacobi-Gauss quadrature points and weights is designed. A detailed convergence analysis of the method is presented in norms L^\infty and L^2_\omega proving spectral accuracy (exponential convergence) for smooth solutions and optimal algebraic convergence even for non-smooth solutions or those with weak singularities at the interval endpoints. Extensive numerical results show that this method is efficient and accurate. Furthermore, the theoretically predicted convergence rate is clearly observed for various problems. In a nutshell, this research demonstrates that spectral methods with a Jacobi basis are a powerful, accurate, and theoretically well-founded tool for solving complex problems such as high-order differential equations and fractional integro-differential equations.