This below is the abstract. The full text, longer than 18000 characters, is attached as a doc. Scientists are asked to describe and understand the complex behaviour of natural processes, and volcanologists provide no exception. Volcanology has moved from an originally descriptive and qualitative investigation to a more quantitative detection and understanding of the processes occurring within volcanoes. This development has allowed reaching an unprecedented understanding of countless aspects of the volcano factory, on magma generation, transport, emplacement and eruption. However, despite the important achievements, modern volcanology, in both its theoretical and applied manifestations, still has to fully capture and understand several major processes. These processes, in an ideal journey from the deepest to the shallowest portions of the volcano factory, consist of: 1) formation and development of magmatic reservoirs; 2) propagation and arrest of magma; 3) emplacement and outcome of shallower intrusions; 4) volcano unrest; 5) eruption forecasting; 6) erupting conditions; 7) collapsing volcanoes; 8) environmental impact of eruptions. Many of these processes may be unravelled not only by observations on volcanoes on Earth, but also on extraterrestrial volcanoes. The definition and understanding of these eight general processes result in likewise challenges which Frontiers in Volcanology intends to address with any type of contribution, method and scale of investigation within the Solar System. Indeed, these first-order processes or challenges embrace all the disciplines of modern volcanology and, as such, are of interest for the broadest volcanological community. These challenges should not be considered as separate entities. Rather, they should be merged in a holistic vision, where any progress in a sector may impact other sectors. In fact, the ultimate challenge for a scientist is to capture the bigger picture from his/her studies, delivering robust and useful data, as well as general models of wide applicability: and this is also the ultimate commitment for Frontiers in Volcanology.