Cryo-EM is a method for determing the structure of the macromolecules and molecular machines that are responsible for cellular function. Cryo-EM single-particle reconstruction (SPR) is the difficult task of deducing the three-dimensional "density" map of a macromolecular "particle" from a set of very noisy electron-microscope images. SPR fails in the cases that the particles are too small, when the images are too noisy, or when high resolution is sought. It fails because under these conditions the orientation of the particles giving rise to the individual images cannot be determined uniquely. We will develop an SPR algorithm that makes use of Robust Estimation theory to more reliably perform reconstructions from challenging datasets. Our algorithm will maximize a clearly-defined statistical quantity (related to the a posteriori probability) while being resistant to the effects of "outlier" images. We will test the algorithm on simulated datasets and on two experimental datasets from small particles. One of these will be the transferrin-transferrin receptor dataset from the Walz lab which provided a successful SPR by conventional techniques;the other will be data acquired from solubilized Slack (hslo2.1) ion channels, for which conventional SPR has to date not been successful. Discerning the structure (detailed shape) of the molecular machines of cells is essential to the understanding of their function, and to the development of therapeutic interventions including drugs that affect their function. Singe-particle cryo-EM is a powerful and flexible method for the determination of structures, but in many cases it fails to provide results, or the results it provides are incorrect. We seek to increase the reliability and usefulness of this method.