Potential energy curves, energy parameters, and spectroscopic values for the X (2)Sigma(+), A (2)Pi, B (2)Sigma(+), a (4)Pi, and b (4)Sigma(+) states of CaH have been calculated using the multireference configuration interaction and coupled cluster levels of theory, while employing quantitative basis sets (of augmented quintuple-zeta quality) and taking also into account core/valence correlation and one-electron relativistic effects. For the ground (X (2)Sigma(+)) and the first two following excited states (A (2)Pi, B (2)Sigma(+)) of CaH, the permanent electric dipole moments have been calculated. Our best finite field dipole moment of the A (2)Pi state of 2.425 D (upsilon = 0) is in very good agreement with the experimental literature value of 2.372(12) D. However, a discrepancy is observed in the dipole moment of the X (2)Sigma(+) state. Our most extensive calculation gives mu = 2.623 D (upsilon = 0), which is considerably smaller than the experimental value of mu = 2.94(16) D (upsilon = 0). Small van der Waals minima were found for both "repulsive" quartet states. Spectroscopic constants and energy parameters for all states are in remarkable agreement with available experimental values.