Abstract. In recent years, several schemes for laser-driven Fast Ignition (FI) of Inertial Confimenement Fusion targets have been proposed. In all schemes, a key element is the conversion of the energy of a Petawatt laser pulse into a beam of strongly relativistic electrons and the transport of the latter into a dense plasma or a solid target. The electron beam may either drive ignition directly or be used to accelerate a proton beam which is turn used to ignite. Both ignition scenarios involve a number of physical processes which are widely unexplored and challenging for theory and simulation. In this contribution, we present theoretical and numerical investigations of several fundamental issues of relevance to FI from the stage of electron generation and transport to that of proton energy deposition, including electron beam instabilities, electron transport in solid-density plasma, proton transport in the coronal plasma, and requirements for proton beam driven ignition.