In this review (containing 185 references) we discuss the vibration-rotational spectroscopy arising from one-photon electric-dipole moment transitions of diatomic molecules in the1∑ state. After a brief survey of the physical effects underlying the interaction of two atoms, we invoke the customary Born-Oppenheimer separation of electronic and nuclear motions in the molecule and introduce the concept of the potential-energy function. We discuss several model functions for the potential energy and their relationship to a general power-series representation, first treated by Dunham, that provides the framework for the analysis of accurate experimental data on transition frequencies. The WKB method of obtaining the term values of energy as functions of the Dunham potential-energy coefficients is described in detail and new results generated by computer algebra are presented. An analytic algorithm for obtaining the wavefunctions corresponding to the Dunham potential-energy function is reviewed and extended. One can use these wavefunctions to derive expressions for expectation values and matrix elements of powers of the reduced displacement from equilibrium; we also discuss several alternative methods, not involving the wavefunctions directly. The use of these matrix elements in the extraction of the dipole-moment function from experimental data on spectral intensities is illustrated by explicit calculations for HC1. In the final section we consider deviations from the Born-Oppenheimer approximation for both the potential-energy and dipole-moment functions, as well as other limitations of the Dunham formalism. Finally we mention briefly some recent trends and developments that we expect to become increasingly important as experimental sophistication advances.