The ultrafast pulses from future X-ray free-electron lasers may enable imaging of non-periodic objects at near-atomic resolution. These objects could include single macromolecules, protein complexes, or virus particles, and the method will be particularly valuable to determine the structures of proteins that cannot be crystallized. The specimen would be completely destroyed by the pulse, but that destruction will only happen after the pulse has passed through the object. The scattering pattern from the object will therefore give structural information about the undamaged object. To address the many challenges that we face in attempting molecular diffraction, we have been developing experimental methods at the FLASH free-electron laser at DESY in Hamburg. We have reconstructed images from single-pulse ultrafast diffraction patterns. These show no evidence of the effects of the FEL pulse on the structure even though the object was completely vaporized by the intense pulse. We also performed quantitative measurements of the explosion of test particles in the focused FEL pulse, by recording their diffraction patterns. No motion occurred during the pulse and we followed the evolution of the explosion with a novel holographic time-resolved technique. Our results confirm the basic principles of flash imaging and lend great confidence to achieving molecular imaging at future short-wavelength X-ray FELs.