Already in 1968 one recognized that the transmission electron microscope could be used in a tomographic setting as a tool for structure determination of macromolecules. However, its usage in mainstream structural biology has been limited and the reason is mostly due to the incomplete data problems that leads to severe ill-posedness of the inverse problem. Despite these problems its importance is beginning to increase, especially in drug discovery. In order to understand the difficulties of electron tomography one needs to properly formulate the forward problem that models the measured intensity in the microscope. The electron-specimen interaction is modelled as a diffraction tomography problem and the picture is completed by adding a description of the optical system of the transmission electron microscope. For weakly scattering specimens one can further simplify the forward model by employing the first order Born approximation which enables us to explicitly express the forward operator in terms of the propagation operator from diffraction tomography acting on the specimen convolved with a point spread function, derived from the optics in the microscope. We next turn to the algorithmic and mathematical difficulties that one faces in dealing with the resulting inverse problem, especially the incomplete data problems that leads to severe ill-posedness. Even though we briefly mention single particle methods, our focus is will be on electron tomography of general weakly scattering specimens and we mention some of the progress that has been made in the field. Finally, if time permits, we provide some examples of reconstructions from electron tomography and demonstrate some of the biological interpretations that one can make.