The paper explores the theoretical background of highly sophisticated mathematical models of lightning strike development in freestanding structures: these are the so-called leader inception and leader progression models. Special attention is paid to the following models: Eriksson’s model, Rizk’s model, Dellera and Garbagnatija’s model, and Becerra and Cooraya’s model. A comparison of the aforementioned mathematical models is made and detailed comments are given on the theoretical assumptions on which the models are based.