3D Zernike functions are defined and used for the reconstruction of precipitate shapes. These functions are orthogonal over the unit ball and allow for an arbitrary shape, scaled to fit inside an embedding sphere, to be decomposed into 3D harmonics. Explicit expressions are given for the general Zernike moments, correcting typographical errors in the literature. Explicit expressions of the Zernike moments for the ellipsoid and the cube are given. The 3D Zernike functions and moments are applied to the reconstruction of γ' precipitate shapes in two Ni-based superalloys, one with nearly cuboidal precipitate shapes, and one with more complex dendritic shapes.