L-splines -- Generalizations of L-splines -- Interpolation and approximation results for piece-wise-polynomials in higher dimensions -- Rayleigh-Ritz-Galerkin method for nonlinear boundary values -- Fourier analysis -- Least squares methods -- Eigenvalue problems -- Parabolic problems -- Chebyshev semidiscrete approximations for linear parabolic problems. The purpose of these lecture notes is to survey in part the enormously expanding literature on the numerical approximation of solutions of elliptic boundary value problems by means of variational and finite element methods. Surveying this area will, as we shall see, require almost constant application of results and techniques from functional analysis and approximation theory to the field of numerical analysis, and it is our hope that the material presented here will serve to stimulate further activity which will strengthen the ties already connecting these fields. Although our primary interest will concern the numerical approximation of elliptic boundary value problems, the methods to be described lend themselves as well rather naturally to discussions concerning eigenvalue problems and initial value problems, such as the heat equation. On the negative side, it is unfortunate that almost nothing will be said here about scientific computing, i.e., the real problems of implementation of such mathematical theories to working programs on high-speed computers, and the numerical experience which has already been gained on such problems. Fortunately, scientific computing is one of the key points of the monograph by Professor Garrett Birkhoff, and we are grateful to be able to refer the reader to this work. The intent of these lecture notes is to make each portion of the notes roughly independent of the remaining material. This is why the references used in each of the nine chapters are compiled separately at the end of each chapter.