Often a crystallographer obtains an electron density map which shows only part of the structure. In such cases, the phasing of the trial model is poor enough that the electron density map may show peaks in some of the atomic positions, but other atomic positions are not visible. There may also be extraneous peaks present which are not due to atomic positions. A method for determination of crystal structures that have resisted solution through normal crystallographic methods has been developed. PHASER is a series of FORTRAN programs which aids in the structure solution of poorly phased electron density maps by refining the crystallographic phases. It facilitates the refinement of such poorly phased electron density maps for difficult structures which might otherwise not be solvable. The trial model, which serves as the starting point for the phase refinement, may be acquired by several routes such as direct methods or Patterson methods. Modifications are made to the reverse transform process based on several assumptions. First, the starting electron density map is modified based on the fact that physically the electron density map must be non-negative at all points. In practice a small positive cutoff is used. A reverse Fourier transform is computed based on the modified electron density map. Secondly, we assume that a better electron density map will result by using the observed magnitudes of the structure factors combined with the phases calculated in the reverse transform. After convergence has been reached, more atomic positions and less extraneous peaks are observed in the refined electron density map;The starting model need not be very large to achieve success with PHASER; successful phase refinement has been achieved with a starting model that consists of only 5 percent of the total scattering power of the full molecule;The second part of the thesis discusses three crystal structure determinations: a porphyrin compound containing two metal atoms, PtCuCl2N8O2C56H52; a cage structure in which two titanium atoms are interconnected via oxygen atoms, Ti2S2O6N2C18H38; and a heterocycle, C20H27OBr, for which the optical isomer was determined. Several other structure solutions are discussed in the first section with respect to the application of PHASER.