We present new calculations of limb and gravity-darkening coefficients to be used as input in many fields of stellar physics such as synthetic light curves of double-lined eclipsing binaries and planetary transits, studies of stellar diameters or line profiles in rotating stars. We compute the limb-darkening coefficients specifically for the photometric system of the satellite MOST (Microvariability and Oscillations in STars). All computations were performed by adopting the least-square method, but for completeness we also performed calculations for the linear and bi-parametric approaches by adopting the flux conservation method. The passband gravity-darkening coefficients y({lambda}) were computed by adopting a more general differential equation, which also takes the effects of convection into account. We used two stellar atmosphere models: ATLAS (plane-parallel) and PHOENIX (spherical and quasi-spherical). We adopted six laws to describe the specific intensity distribution: linear, quadratic, square root, logarithmic, exponential, and a more general one with four terms. The covered ranges of T_eff_, log g, metallicities, and microturbulent velocities are [1500-50000K, 0-5.5,-5.0-1.0, 0-8km/s], respectively.