X-ray diffractive imaging is set apart from other high-resolution imaging techniques (e.g. scanning electron or atomic force microscopy) for its high penetration depth, which enables tomographic 3D imaging of thick samples and buried structures. Furthermore, using short x-ray pulses, it enables the capability to take ultrafast snapshots, giving a unique opportunity to probe nanoscale dynamics at femtosecond time scales. In this thesis we present improvements to phase retrieval algorithms, assess their performance through numerical simulations, and develop new methods for both imaging and wavefront measurement.;Building on the original work by Faulkner and Rodenburg, we developed an improved reconstruction algorithm for phase retrieval with transverse translations of the object relative to the illumination beam. Based on gradient-based nonlinear optimization, this algorithm is capable of estimating the object, and at the same time refining the initial knowledge of the incident illumination and the object translations. The advantages of this algorithm over the original iterative transform approach are shown through numerical simulations.;Phase retrieval has already shown substantial success in wavefront sensing at optical wavelengths. Although in principle the algorithms can be used at any wavelength, in practice the focus-diversity mechanism that makes optical phase retrieval robust is not practical to implement for x-rays. In this thesis we also describe the novel application of phase retrieval with transverse translations to the problem of x-ray wavefront sensing. This approach allows the characterization of the complex-valued x-ray field in-situ and at-wavelength and has several practical and algorithmic advantages over conventional focused beam measurement techniques. A few of these advantages include improved robustness through diverse measurements, reconstruction from far-field intensity measurements only, and significant relaxation of experimental requirements over other beam characterization approaches. Furthermore, we show that a one-dimensional version of this technique can be used to characterize an x-ray line focus produced by a cylindrical focusing element. We provide experimental demonstrations of the latter at hard x-ray wavelengths, where we have characterized the beams focused by a kinoform lens and an elliptical mirror. In both experiments the reconstructions exhibited good agreement with independent measurements, and in the latter a small mirror misalignment was inferred from the phase retrieval reconstruction. These experiments pave the way for the application of robust phase retrieval algorithms for in-situ alignment and performance characterization of x-ray optics for nanofocusing. We also present a study on how transverse translations help with the well-known uniqueness problem of one-dimensional phase retrieval.;We also present a novel method for x-ray holography that is capable of reconstructing an image using an off-axis extended reference in a non-iterative computation, greatly generalizing an earlier approach by Podorov et al. The approach, based on the numerical application of derivatives on the field autocorrelation, was developed from first mathematical principles. We conducted a thorough theoretical study to develop technical and intuitive understanding of this technique and derived sufficient separation conditions required for an artifact-free reconstruction. We studied the effects of missing information in the Fourier domain, and of an imperfect reference, and we provide a signal-to-noise ratio comparison with the more traditional approach of Fourier transform holography. We demonstrated this new holographic approach through proof-of-principle optical experiments and later experimentally at soft x-ray wavelengths, where we compared its performance to Fourier transform holography, iterative phase retrieval and state-of-the-art zone-plate x-ray imaging techniques (scanning and full-field). Finally, we present a demonstration of the technique using a single 20 fs pulse from a high-harmonic table-top source. Holography with an extended reference is shown to provide fast, good quality images that are robust to noise and artifacts that arise from missing information due to a beam stop. (Abstract shortened by UMI.)
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