The idea of solving unknown crystal structures from experimental electron-diffraction intensities and high-resolution electron micrographs has remained a controversial topic in the 60 year history of electron crystallography. In this review it will be shown that the application of modern direct phasing techniques, familiar to X-ray crystallographers, has decisively proven that such ab initio determinations are, in fact, possible. This statement does not, by any means, refute the existence of the several significant scattering perturbations identified by diffraction physicists. Rather, it does affirm that experimental parameters can be controlled to ensure that a `quasi-kinematical' data set can be collected from many types of specimens. Numerous applications have been made to various types of specimens, ranging from small organics to proteins, and also some inorganic materials. While electron crystallography may not be the optimal means for determining accurate bonding parameters, it is often the method of choice when only microcrystalline specimens are available.