The goal in this chapter is to present a theory of one-dimensional grid generation on the line to illustrate some of the basic concepts involved in grid-generation algorithms. A major goal of one-dimensional grid-generation algorithms is to create grids with specified distances between the grid points, thus the chapter begins with a simple and intuitive geometric argument to show how to produce second-order differential equations whose solutions are one-dimensional grids with specified segment lengths. This discussion provides a firm foundation for developing solution-adaptive grid generators. The chapter discusses a simple discrete minimization problem, and then extends to the continuum to produce a variational algorithm for one-dimensional grid generation. It reviews basic results in the classical calculus of variations. The chapter closes with a discussion of one-dimensional solution-adaptive methods.