When a molecule is subject to a strong laser field, there is a probability that an electron can escape, even though the electrons are bound by a large po- tential barrier. This is possible because electrons are quantum mechanical in nature, and they are therefore able to tunnel through potential barriers, an ability classical particles do not possess. Tunnelling is a fundamental quantum mechanical process, a process that is distinctly non-classical, so solving this tunnelling problem is not only relevant for molecular physics, but also for quantum theory in general. In this dissertation the theory of tunneling ionization of molecules is presented and the results of numerical calculations are shown. One perhaps surprising result is, that the frequently used Born-Oppenheimer approximation breaks down for weak fields when describing tunneling ionization. An analytic theory applicable in the weak-field limit which supplements the Born-Oppenheimer approximation is also presented.