This chapter systematically presents the Legendre and associated Legendre functions and spherical harmonics. These functions are defined by deriving a finite solution of Laplace’s equation in the spherical coordinate system based on Fourier’s method of variable separation. Several subjects usually not found in introductory texts are covered. These include some extra orthogonality formulae, an asymptotic expression of the Legendre function, the closed-form formulae of some Legendre sums, the isotropic smoothing of functions defined on a sphere, the expression of a three-dimensional scalar field in spherical harmonics, and the decomposition of a three-dimensional vector field into scalar fields followed with their expressions in spherical harmonics. As an application, the partial differential equations governing the elastic deformation of the Earth are reduced to a system of ordinary differential equations with the distance to the Earth center as the variable. Finally, the discrete spherical transform, together with the discrete Fourier transform and discrete Chebyshev polynomial transform, is presented. As a supplement, a numerical method for solving differential equations, called Chebyshev collocation method, is presented, which is particularly suitable for handling singularities of the equations at a boundary, which is the case of the system of ordinary differential equations mentioned.