Energies of the ground and excited states of excitons in GaAs/AlGaAs and InGaAs/GaAs finite square quantum wells (QWs) of various widths are calculated. This is achieved by studying the three-dimensional Schrodinger equation for the exciton in a QW and, in particular, by determining the lower energy boundary of the continuous spectrum of the corresponding differential operator. The eigenvalue problem for the Schrodinger equation is solved numerically by the finite-difference method properly taking into account discontinuities of the material parameters at the interfaces of the QW. The calculated bound states of electron-hole pairs are classified based on the types of their dominant in-plane and quantum-confinement one-dimensional functions of the wave function factorized form. A dependence of energy levels on a QW width as a parameter is thoroughly studied for widths up to 100 nm. The accurate radiative decay rates for calculated s-like exciton states are also obtained. Calculated energy spectra are confronted with the experimental reflectance spectra measured for high-quality InGaAs/GaAs heterostructures with QWs. The ground and, at least, a few excited states of the heavy-hole exciton in QW are identified in the experimental spectra.