The vibrational states experienced by the active components of a drilling assembly such as that found in the oil or gas industry are discussed in the context of an integrated mathematical model. The work is motivated by the need to understand the complex vibrational states that such a system can exhibit in order to better control their constructive and destructive potential. The model is expressed in terms of six continuous independent degrees of freedom. Three locate the position of the centroid of the drill-string in space and three permit the dynamical state of the drill-string to be expressed in terms of ยฏexural, torsional and shear strain, together with dilation and stretch. By supplementing the model with appropriate constitutive relations that relate these strains to bending and twisting couples together with shear and compression forces it can fully accommodate the modes of vibration that are traditionally associated with the motion of drill-strings in both straight and curved boreholes discussed in the engineering literature. These include axial motion along the length of the drill-string, torsional or rotational motion and transverse or lateral motion. Attention is given to the boundary conditions appropriate for an active drill-string and BHA stabiliser including an account of frictional simulations at the rock-interface, cutter simulations for dierent types of drill-bit and interactions between the bore cavity and the drill-string. The model is used to discuss the stability of axisymmetric drill-string conยฎgurations in vertical boreholes under both coupled torsional, axial and lateral perturbations as well as general non-perturbative coupled vibrational states under extreme conditions of lateral whirl.