The detection of the first exoplanet paved the way for the era of transit-photometry space missions with revolutionary photometric precision, whose aim is to discover new exoplanetary systems around different types of stars. With this high precision, it is possible to derive the radii of exoplanets very accurately, which is crucial for constraining their type and composition. However, it requires an accurate description of their host stars, especially their center-to-limb variation of intensities (so-called limb darkening) as it affects the planet-to-star radius ratio determination. We aim to improve the accuracy of limb-darkening calculations for stars with a wide range of fundamental parameters. We used the recently developed 1D Merged Parallelized Simplified ATLAS (MPS-ATLAS) code to compute model atmosphere structures and to synthesize stellar limb darkening on a very fine grid of stellar parameters. For the computations, we utilized the most accurate information on chemical element abundances and mixing-length parameters, including convective overshoot. The stellar limb darkening was fitted using the two most accurate limb darkening laws: the power-2 and 4-parameter nonlinear laws. We present a new extensive library of stellar model atmospheric structures, the synthesized stellar limb darkening curves, and the coefficients of parameterized limb-darkening laws on a very fine grid of stellar parameters in the Kepler, TESS, CHEOPS, and PLATO passbands. The fine grid allows the sizable errors, introduced by the need to interpolate, to be overcome. Our computations of solar limb darkening are in a good agreement with available solar measurements at different view angles and wavelengths. Our computations of stellar limb darkening agree well with available measurements of Kepler stars. A new grid of stellar model structures, limb darkening, and their fitted coefficients in different broad passbands are provided.