  The semi-classical Bloch-Boltzmann theory is at the heart of our
understanding of conduction in solids, ranging from metals to semi-conductors.
Physical systems that are beyond the range of applicability of this theory are
thus of fundamental interest. This is the case of disordered systems which
present quantum interferences in the diffusive regime, i.e. Anderson
localization effects. It appears that in quasicrystals and related complex
metallic alloys another type of breakdown of the semi-classical Bloch-Boltzmann
theory operates. This type of quantum transport is related to the specific
propagation mode of electrons in these systems. We develop a theory of quantum
transport that applies to a normal ballistic law but also to these specific
diffusion laws. As we show phenomenological models based on this theory
describe correctly the experimental transport properties. Ab-initio
calculations performed on approximants confirm also the validity of this
anomalous quantum diffusion scheme. Although the present chapter focuses on
electrons in quasicrystals and complex metallic alloys, the concept that are
developed here can be useful for phonons in these systems. There is also a deep
analogy between the type of quantum transport described here and the conduction
properties of other systems where charge carriers are also slow, such as some
heavy fermions or polaronic systems.
