In this PhD thesis, the compressible fluid dynamics of high-speed impactof a spherical liquid droplet on a rigid substrate is investigated. The impact phenomenonis characterised by the compression of the liquid adjacent to the targetsurface, whereas the rest of the liquid droplet remains unaware of the impact. Initially,the area of compressed liquid is assumed to be bounded by a shock envelope,which propagates both laterally and upwards into the bulk of the motionlessliquid. Utilizing a high-resolution axisymmetric solver for the Euler equations, itis shown that the compressibility of the liquid medium plays a dominant role inthe evolution of the phenomenon. Compression of the liquid in a zone defined bya shock wave envelope, lateral jetting of very high velocity and expansion wavesin the bulk of the medium are the most important mechanisms identified, simulatedand discussed.During the first phase of impact, all wave propagation velocities aresmaller than the contact line velocity, thus the shock wave remains attached to thelatter. At a certain point, the radial velocity of the contact line decreases below theshock velocity and the shock wave overtakes the contact line, starting to travelalong the droplet free surface. The resulting high pressure difference across thefree surface at the contact line region triggers an eruption of intense lateral jetting.The shock wave propagates along the free surface of the droplet and it is reflectedinto the bulk of the liquid as an expansion wave. The development of pressure anddensity in the compressed area are numerically calculated using a front trackingmethod. The exact position of the shock envelope is computed and both onset andmagnitude of jetting are determined, showing the emergence of liquid jets of veryhigh velocity (up to 6000 m/s). Computationally obtained jetting times are validatedagainst analytical predictions. Comparisons of computationally obtainedjetting inception times with analytic results show that agreement improves significantly if the radial motion of the liquid in the compressed area is taken intoaccount.An analytical model of the impact process is also developed and comparedto the axisymmetric numerical solution of the inviscid flow equations.Unlike the traditional linear model - which considers all wave propagation velocitiesto be constant and equal to the speed of sound, the developed model predictsthe exact flow state in the compressed region by accommodating the real equationof state. It is shown that the often employed assumption that the compressed areais separated from the liquid bulk by a single shock wave attached to the contactline, breaks down and results in an anomaly. This anomaly emerges substantiallyprior to the time when the shock wave departs from the contact line, initiating lateralliquid jetting. Due to the lack of more sophisticated mathematical models, thistended to be neglected in most works on high speed droplet impact, even thoughit is essential for the proper understanding of the pertinent physics. It is proven thatthe presence of a multiple-wave structure (instead of a single shock wave) at thecontact line region resolves the aforementioned anomaly. The occurrence of thismore complex multiple wave structure is also supported by the numerical results.Based on the developed analytical model, a parametric representation ofthe shock envelope surface is established, showing a substantial improvementwith respect to previous linear model, when validated against numerical findings.In the final part of the thesis, the assumption of a multiple wave structurewhich removes the above mentioned anomaly is underpinned with an analyticalproof showing that such a structure is indeed a physically acceptable solution.