The theoretical model which is widely used to describe the quantum mechanical tunnelling process, namely the effective mass theory of tunnelling, is presented along with some of its limitations. The paper then describes experiments by the author and colleagues to test effective mass theory. Experimental investigations of the effect of hydrostatic pressure (up to 15 kilobar) and magnetic field (up to 11T) on the low temperature J(V) characteristics of single barrier n+GaAs/(AIGa)As/n−GaAs/n+GaAs tunnelling structures are described. The pressure dependence is accurately described by the effective mass/WKB model up to 10 kilobar. At higher pressure a rapid fall in the carrier concentration in the n+GaAs contact is observed due to pressure-induced trapping of electrons on defect levels, probably associated with the L-or X-conduction band minima of GaAs. The reduction of the tunnelling current in an applied magnetic field is discussed in terms of the effect of the diamagnetic energy in increasing the height of the potential barrier. The final section of the paper draws some general conclusionsfrom the theory and the experimental work presented.