This dissertation primarily presents new procedures to analyze the lateral vibrations of rotary oil-well drilling assemblies. The methodology uses random vibrations theory to gain insight into the complicated downhole dynamics; it is also applicable to longitudinal, torsional and coupled vibrations. Furthermore, this dissertation discusses a novel technique to model drilling vibratory excitation mechanisms using digital filters, presents an error estimator for the mode superposition method, introduces drilling fundamentals and includes an extensive literature review on drilling vibrations and related topics. A finite-element system represents the drilling assembly; modal analysis yields its natural frequencies and mode shapes. Additional structural dynamics techniques enable the reduction of the system size and the approximation of the transfer function using the mode superposition method. Furthermore, Hertz's contact law allows considering the well-bore presence whenever the lateral displacement of any node along the drill-string exceeds its pre-assigned clearance. Accounting for contact between the drilling assembly and the formation introduces nonlinearity in the system. Next, the drill-string response to a harmonic excitation omitting and accounting for the effects of the well-bore presence is obtained. Further, auto-regressive moving-average (ARMA) filters permit modeling measurement-while-drilling data and synthesizing time histories compatible with the generated approximate power spectra. The influence of the bit type on excitation mechanisms leads to separate treatments of bottom-hole assemblies with drag bits and those with roller-cone bits. The ARMA-filter-generated time histories are artificial downhole excitations. Used in conjunction with Monte-Carlo simulations, they yield the response of the system to random excitations. The stochastic linearization method is adopted for generating a linear system equivalent, in a statistical sense, to the original nonlinear problem. The solution of the stochastic vibrations problem of the equivalent system is obtained with two solution procedures: the covariance matrix approach in the time-domain, and the spectral matrix technique in the frequency-domain. Numerical studies are used to assess the relative efficiency of the three solution methods. It is hoped that the developed approach will stimulate further interest in the suitability of stochastic dynamics methods for the inherently uncertain environment of oil-well drilling.