The paper gives a complete and consistent analysis of discontinuous phase transitions in lattice models. The authors developed a general method to check this kind of phase transitions via discontinuity of the transition in the associated mean-field theory. In spite of to develop their method the authors use essentially a “magnetic language”, their theory is general to be easily accommodated other type models. As an application the authors consider Potts-type models and a non-diluted \(O(N)\)-nematic liquid-crystal model.