We present new phenomenological optical model potentials (OMPs) for neutrons and protons with incident energies from 1 keV up to 200 MeV, for (near-)spherical nuclides in the mass range 24⩽ A⩽209. They are based on a smooth, unique functional form for the energy dependence of the potential depths, and on physically constrained geometry parameters. For the first time, this enables one to predict basic scattering observables over a broad mass range and over an energy range that covers several orders of magnitude in MeV. Thereby, the necessity of using different OMPs in different energy regions has been removed. Using extensive grid searches and a new computational steering technique, we have obtained optical model parameters for many isotopes separately. We recommend that the resulting, so-called local, optical models be used in theoretical analyses of nuclear data. From these parameterizations, we have also constructed asymmetry-dependent neutron and proton global OMPs that are superior to all other existing phenomenological ones, not only with respect to the description of observables, but also as they cover larger mass and energy ranges. These (nucleon) global OMPs, we believe, may be used with some confidence in other studies whenever one of our local OMPs does not exist. To constrain our parameterization as much as possible and to assess the performance of our OMPs, we have compared our calculated results with an extensive experimental data set. This data set includes average resonance parameters, total and non-elastic cross sections, elastic scattering angular distributions and analyzing powers. The numerous local OMPs we have obtained allow us to disentangle asymmetry, Coulomb correction and mass-dependent components of our global OMPs.