A first principles method for the computation of electric field gradients (EFGs) is illustrated for various solids. This scheme is based on self-consistent energy band-structure calculations by the full potential linearized augmented plan wave (FLAPW) method which provides the electronic charge density including all polarization effects. By numerically solving Poisson's equation we obtain the Coulomb potential in a form which allows to compute the EFG directly. Our method is demon-strated for insulators (Cu20), metals (hcp-Zn), supercondutors (YBa2Cu3Ö7) and molecular crystals (Cl2, Br2, I2).