The era of easily acquiring oil and gas is over. Now, to extract these resources, deep vertical and curvilinear, off-shore and ground-based boreholes are drilled in inclement climate conditions and in complex heterogeneous tectonic rocks. Additional novelties have also been assimilated into these technologies by the shale revolution. The techniques and methods of borehole drivage are developing so fast that scientific understanding and substantiation have no time to refine them. Therefore, as a rule, different unpredictable emergency effects and overall failures accompany these procedures. The gravest of these are buckling of the drill strings, their resonant vibrations, their sticking during dragging and twisting, the self-triggering of torsional relaxation vibration, and the self-excitation of forward, backward, and superfast bit whirlings. To analyze these phenomena, this volume elaborates new mathematic models and numerical methods. On the basis of these, a special software was created, which allowed new results concerning the peculiarities of the affected phenomena to be achieved. The book will be of interest to borehole designers and specialists in mechanics and applied mathematics.