This book presents a new approach to analyze quantum mechanical tunnelling of particles across potential barriers. The conventional concepts of this phenomenon, which are based on a time-in-dependent or a time-dependent perturbation approach are inadequate in furnishing explanations to a number of effects, e.g. (i) the limit of resolution of a field emission macroscope (FEM), (ii) Esaki integral, representing the I-V characteristics of solid-state junctions, (iii) Josephson effect, (iv) tunnelling time, (v) tunnelling current density etc. The new analysis presented here not only provides adequate explanations to all the above mentioned effects but also furnishes an appropriate expression for the tunnelling current density which yields results closer to experimentally observed values.