Let X be a compact Kähler manifold. The so-called Kodaira problem asks whether X has arbitrarily small deformations to some projective varieties. While Kodaira proved that such deformations always exist for surfaces, starting from dimension 4 there are examples constructed by Voisin which answer the Kodaira problem in the negative. In this talk, we will focus on threefolds and explain our positive solution to the Kodaira problem in this situation.