In this thesis, we have examined two categories of fractional calculus problems. These two categories are linear and nonlinear fractional ordinary differential equations. For this purpose, first of all, some necessary definitions and the-orems such as the introduction of the weight function and continuous inner product and orthogonality and orthogonal polynomials, Gaussian quadrature, and an introduction to three special functions (gamma, beta, and Mittag-Leffler) that play the most important role in the theory of fractional calculus and fractional differential equations are given.