Coherent Diffractive Imaging (CDI) has become an increasingly popular frame work in which to solve the classic phase problem in imaging due to benefits in resolution and the facility of collecting data in this modality. In particular, there is considerable interest in using the short wavelength and high coherence of fourth generation x-ray sources with CDI techniques to phase non-crystalline, or nano-crystalline biomolecular samples. CDI provides an opportunity to determine the structure of proteins and other biological samples which are unable to be phased with the standard techniques of protein crystallography, typically due to lack of adequate crystalline samples. Methods of non-crystalline phase retrieval are legion, however many suffer from limitations in resolution or the inability to recover phase fields containing a pathological singularity. Wavefields containing phase singularities are common in optical fields. The creation of an x-ray wavefield containing a pathological phase singularity is demonstrated. In this thesis a form of CDI termed astigmatic diffraction is presented, that is able tophase uniquely this class of wavefield. This is achieved by illuminating the sample with beams containing known phase curvature. The theory of the method and simulations of its application to a nano-crystalline biomolecule are presented. The experimental recovery of the direction of the phase gradient of a sample illuminated with coherent x-rays produced by a synchrotron source is shown to verify this method.