The confluence of three technical advances in the past decade has led to a breakthrough in high resolution imaging with x-rays. The advent of brilliant third-generation synchrotron x-ray sources enabled pioneering advances in coherent x-ray experiments. The availability of highly sensitive megapixel x-ray cameras made it possible to record x-ray diffraction patterns in exquisite detail. But while even a candle produces some coherent light and photographic film can be used to record diffraction patterns, the most significant recent advance is development of computational phase retrieval algorithms for recovering images from the recorded diffraction data. Interest by the x-ray imaging community in the basic algorithms, conceived in the 1970s and 1980s for optical and electron imaging, surged when it became evident that enough coherent x-ray photons could be detected to image objects at unprecedented resolution. X-ray coherent diffractive imaging has already made an impact in the condensed matter and life sciences. Fueled by a new generation of ultra-bright x-ray laser sources and fast x-ray detectors, it is poised to revolutionize our understanding of the relationships between structure and function in materials and biological systems, for example, how electron spins interact on the femtosecond scale in magnetic devices, and how sub-cellular organelles function within biological cells. This capability offers tremendous opportunity yet it exposes fresh challenges that currently limit progress. Much work lies ahead to develop efficient algorithms for analyzing complex and noisy diffraction data sets, and practical tools for effective coordination between investigators and analysis efforts. This talk reviews the current status of the field and discusses how algorithmic advances can enable science by coherent diffractive imaging.