We address the problem of phase inpainting, i.e. the reconstruction of partially-missing phases in linear measurements. We thus aim at reconstructing missing phases of some complex coefficients assuming that the phases of the other coefficients as well as the modulus of all coefficients are known. The mathematical formulation of the inverse problem is first described and then, three methods are proposed: a first one based on the well known Griffin and Lim algorithm and two other ones based on positive semidefinite programming (SDP) optimization methods namely PhaseLift and PhaseCut, that are extended to the case of partial phase knowledge. The three derived algorithms are tested with measurements from a short-time Fourier transform (STFT) in two situations: the case where the missing data are distributed uniformly and indepedently at random and the case where they constitute holes with a given width. Results show that the knowledge of a subset of phases contributes to improve the signal reconstruction and to shorten the convergence of the optimization process.