Drilling systems exhibit complex behaviors characterized by strong oscillations, prompting researchers to study their stability and develop control techniques. The nonlinear interaction at the bit-rock interface is a major problem when trying to model the dynamics of the drill string. This is what causes the vibrations, especially the coupled axial-torsional vibrations. This area has garnered substantial interest, leading to the development of various bit-rock interaction models, ranging from simplified to complex, incorporating both uni-directional and bi-directional couplings. This paper brings together the existing bit-rock interaction models that are used to simulate coupled axial and torsional vibrations in drilling systems, particularly at the bit-rock interface. It includes a detailed examination of the assumptions and simplifications employed in these analytical studies.