1 Introduction  
2 Decomposing the lepton number breaking operator $\mathcal{O}_{9}$  
3 UV completions of the $\mathcal{O}_{9}$ operator and their phenomenology  
3.1 Class 1  
3.2 Class 2  
4 Neutrino mixing and neutrinoless double $\beta$-decay  
4.1 Class 1  
4.2 Class 2  
5 Conclusions