High peak power lasers have been used for years for ablating matter. The most relevant application of this process is laser marking. Marking meets the demands of applications although the quality of ablation has potential to be further improved. However, the qualitative results of the ablation process especially for highly efficient removal of matter in the liquid phase like drilling have not met the standards of alternative processes—the latter is only the case in niches. On the other hand, the ablation by ultrafast lasers in the pulse regime of ps or below, which might meet the quality demands in terms of geometric precision, was too slow for economically feasible application because of the lack of average power. In fact, both process domains have been developed substantially and thus lead to a technological level which make them ready for industrial innovations. In a series of three articles on laser drilling—from fundamentals to application technology—the results of more than a decade of research and development are summarized with the purpose of displaying the bright application future of this laser process. This present part I deals with fundamentals, modeling, and simulation of laser drilling. Part II covers processing techniques, whereas part III is dedicated to systems and application technology. Fundamentals, modeling, and simulation: Theoretical analysis of the process from fs- to μs-pulses involves three inputs: numerical simulation, relevant analytic modeling, and as an important input for understanding, process analysis. The reduction of the models guided by experimental input leads to descriptions and knowledge of the process, which allows for strategic improvement of the applicability. As a consequence, process strategies can be derived, meeting the challenges of the application related to shape and accuracy of the surface free of recast as well as the economical demand for high speed processing. The domains of “cold ablation,” “hot ablation,” and “melt expulsion” are differentiated. Especially, the formation of recast up to closure of the drill is quantified. Tailoring the process parameters toward the individual application according to the know-how reached by the state of the art modeling and simulation leads to sound innovations and shorter innovation cycles.