In this dissertation, the process of recovering structure from sparse data is discussed. Specifically, studies are undertaken for the case of X-ray imaging with weak signal. The limit of interest is when each image (or data frame) is so sparsely populated with X-ray photons that the structure of the object is not discernable. These techniques will be required to perform single molecule imaging using X-ray free electron lasers and serial microcrystallography experiments at synchrotron sources. An overview of the problem and the reconstruction algorithm is given in Chapter 1. In Chapters 2, 3 and 4, three different experiments are discussed, each with a different imaging geometry. All three experiments have the same property in that there are many data frames all of whom are very sparsely occupied. In Chapter 2, a shadowgraphy experiment is performed with a randomly rotated mask whose projected shadow on the detector is reconstructed from a large number of images with a signal level as low as 2.5 photons per data frame. In Chapter 3, using a computed tomography (CT) setup with unrecorded orientations, the 3D structure of a plastic figure is reconstructed. And finally, in Chapter 4, a weak beam is used to illuminate a crystal and the sparse diffraction pattern is measured. The successful reconstruction of the 3D intensity distribution shows promise for the possibility of serial microcrystallography [5] at conventional syn- chrotron sources. Appendix A contains a more detailed discussion of the reconstruction process for the crystallography experiment in Chapter 4. This record is made to enable other parties to reproduce the results as well as to document the intuition used in choosing various reconstruction parameters.