Popularity of the finite element method is such that an astute lecturer or author may increase his audience by choice of a title like "Finite Elements and ," inserting his topic in the blank space, no matter how remotely connected with finite element methods. The title of this book is, nevertheless, precisely the subject of this book. Fundamental to any finite element computation is the definition of an approximation space over a collection of elements. A basis function is associated with each element node so that the approximation within the element is determined by the nodal values. Polynomial basis functions have been widely used, and convergence theory for continuous patchwork polynomial approximation has been developed to a high degree of mathematical sophistication. Elements over which polynomial basis functions apply, within restrictions imposed for rigorous theoretical foundations, are extremely limited. In two space dimensions, for example, triangles and parallelograms are admissible. Isoparametric coordinates enable use of a larger class of three- and four-sided elements that may have parabolic as well as straight sides. Despite limitations on element geometry, polynomial and isoparametric basis functions seem adequate for finite element computations of current concern. Why then do we seek alternatives?