A shaped charge is an explosive device used to penetrate thick targets using a high velocity jet. A typical shaped charge contains explosive material behind a conical hollow. The hollow is lined with a compliant material, such as copper. Extremely high stresses caused by the detonation of the explosive have a focusing effect on the liner, turning it into a long, slender, stretching jet with a tip speed of up to 12km/s. A mathematical model for the penetration of this jet into a solid target is developed with the goal of accurately predicting the resulting crater depth and diameter. The model initially couples fluid dynamics in the jet with elastic-plastic solid mechanics in the target. Far away from the tip, the high aspect ratio is exploited to reduce the dimensionality of the problem by using slender body theory. In doing so, a novel system of partial differential equations for the free-boundaries between fluid, plastic and elastic regions and for the velocity potential of the jet is obtained. In order to gain intuition, the paradigm expansion-contraction of a circular cavity under applied pressure is considered. This yields the interesting possibility of residual stresses and displacements. Using these ideas, a more realistic penetration model is developed. Plastic flow of the target near the tip of the jet is considered, using a squeeze-film analogy. Models for the flow of the jet in the tip are then proposed, based on simple geometric arguments in the slender region. One particular scaling in the tip leads to the consideration of a two-dimensional paradigm model of a ``filling-flow'' impacting on an obstacle, such as a membrane or beam. Finally, metallurgical analysis and hydrocode runs are presented. Unresolved issues are discussed and suggestions for further work are presented.