The physical processes of field electron emission (FE) and electrostatic field ionization (ESFI) both involve quantum-mechanical (QM) tunnelling. They are two of the (perhaps) seven historical paradigm examples of QM tunnelling. A modern relevance of FE and ESFI is that each forms part of the theory of various modern technologies and scientific techniques. However, there are fundamental quantum aspects of both FE theory and ESFI theory that are not fully understood. This chapter aims to identify some of these fundamental difficulties and to offer some ideas for wider discussion.