  Characterization of the atmospheres of transiting exoplanets relies on
accurate measurements of the extent of the optically thick area of the planet
at multiple wavelengths with a precision $\lesssim$100 parts per million (ppm).
Next-generation instruments onboard the James Webb Space Telescope (JWST) are
expected to achieve $\sim$10 ppm precision for several tens of targets. A
similar precision can be obtained in modelling only if other astrophysical
effects, including the stellar limb-darkening, are accounted for properly. In
this paper, we explore the limits on precision due to the mathematical formulas
currently adopted to approximate the stellar limb-darkening, and to the use of
limb-darkening coefficients obtained either from stellar-atmosphere models or
empirically. We propose a new limb-darkening law with two coefficients,
`power-2', which outperforms other two-coefficient laws adopted in the
literature in most cases, and particularly for cool stars. Empirical
limb-darkening based on two-coefficient formulas can be significantly biased,
even if the light-curve residuals are nearly photon-noise limited. We
demonstrate an optimal strategy to fitting for the four-coefficients
limb-darkening in the visible, using prior information on the exoplanet orbital
parameters to break some of the degeneracies that otherwise would prevent the
convergence of the fit. Infrared observations taken with the James Webb Space
Telescope (JWST) will provide accurate measurements of the exoplanet orbital
parameters with unprecedented precision, which can be used as priors to improve
the stellar limb-darkening characterization, and therefore the inferred
exoplanet parameters, from observations in the visible, such as those taken
with Kepler/K2, JWST, other past and future instruments.
