  It has been recently found that the equations of motion of several
semiclassical systems must take into account anomalous velocity terms arising
from Berry phase contributions. Those terms are for instance responsible for
the spin Hall effect in semiconductors or the gravitational birefringence of
photons propagating in a static gravitational field. Intensive ongoing research
on this subject seems to indicate that actually a broad class of quantum
systems might have their dynamics affected by Berry phase terms. In this
article we review the implication of a new diagonalization method for generic
matrix valued Hamiltonians based on a formal expansion in power of $\hbar$. In
this approach both the diagonal energy operator and dynamical operators which
depend on Berry phase terms and thus form a noncommutative algebra, can be
expanded in power series in \hbar $. Focusing on the semiclassical
approximation, we will see that a large class of quantum systems, ranging from
relativistic Dirac particles in strong external fields to Bloch electrons in
solids have their dynamics radically modified by Berry terms.
