We discuss several ways in which molecular absorption and scattering spectra can be computed ab initio, from the fundamental constants of nature. These methods can be divided into two general categories. In the first, or sequential, type of approach, one first solves the electronic part of the Schrödinger equation in the Born–Oppenheimer approximation, mapping out the potential energy, dipole moment vector (for infrared absorption) and polarizability tensor (for Raman scattering) as functions of nuclear coordinates. Having completed the electronic part of the calculation, one then solves the nuclear part of the problem either classically or quantum mechanically. As an example of the sequential ab initio approach, the infrared and Raman rotational and vibrational‐rotational spectral band contours for the water molecule are computed in the simplest rigid rotor, normal mode approximation. Quantum techniques are used to calculate the necessary potential energy, dipole moment, and polarizability information at the equilibrium geometry. A new quick, accurate, and easy to program classical technique involving no reference to Euler angles or special functions is developed to compute the infrared and Raman band contours for any rigid rotor, including asymmetric tops. A second, or simultaneous, type of ab initio approach is suggested for large systems, particularly those for which normal mode analysis is inappropriate, such as liquids, clusters, or floppy molecules. Then the curse of dimensionality prevents mapping out in advance the complete potential, dipole moment, and polarizability functions over the whole space of nuclear positions of all atoms, and a solution in which the electronic and nuclear parts of the Born–Oppenheimer approximation are simultaneously solved is needed. A quantum force classical trajectory (QFCT) molecular dynamic method, based on linear response theory, is described, in which the forces, dipole moment, and polarizability are computed quantum mechanically, using gradient techniques step by step along a classical trajectory whose path is determined by these quantum forces. We believe the QFCT method to be a more practical ab initio route to spectral band contours for large molecules, clusters, and solutions, and it can be equally applied to equilibrium and nonequilibrium systems. It is pointed out that a similar ab initio QFCT molecular dynamic approach could be used to compute other types of spectra, e.g., electronic absorption, as well as other parameters such as transport properties and thermodynamic functions and their quantum corrections. For parameters not depending on momenta, a parallel ab initio Monte Carlo approach would use electronic energies and other parameters of interest generated quantum mechanically, and ‘‘classical’’ trial moves of the nuclei.