A method is presented for three-dimensional reconstruction from electron micrographs of a specimen containing a disordered collection of identical objects with unknown orientations. All the data from all the images are simultaneously used to obtain an approximately maximum likelihood estimate of the three-dimensional electron density, which is represented as a truncated expansion in a complete orthonormal set of basis functions. Anomalous objects can nevertheless be detected and eliminated. The method remains under statistical control, and a hypothesis test is used to choose the lowest resolution reconstruction that is consistent with the data. Error propagation is quantitatively traced from the micrograph to the reconstructed electron density. Random orientation is not necessary, and prior knowledge of preferred orientation can be used to advantage. Similarly, symmetry in the object is not necessary, but it can be imposed and exploited, if appropriate. Evidence is presented that useful reconstructions can be obtained with only one or two extra tilts from highly disordered specimens, even if the objects are asymmetric. The companion paper discusses in detail the implementation and verification of the method.