Single-particle structure recovery without crystals or radiation damage is a revolutionary possibility offered by X-ray free-electron lasers, but it involves formidable experimental and data-analytical challenges. Many of these difficulties were encountered during the development of cryogenic electron microscopy of biological systems. Electron microscopy of biological entities has now reached a spatial resolution of about 0.3 nm, with a rapidly emerging capability to map discrete and continuous conformational changes and the energy landscapes of biomolecular machines. Nonetheless, single-particle imaging by X-ray free-electron lasers remains important for a range of applications, including the study of large "electron-opaque" objects and time-resolved examination of key biological processes at physiological temperatures. After summarizing the state of the art in the study of structure and conformations by cryogenic electron microscopy, we identify the primary opportunities and challenges facing X-ray-based single-particle approaches, and possible means for circumventing them.