  This work presents an approach towards the representation theory of the braid
groups $B_n$. We focus on finite-dimensional representations over the field of
Laurent series which can be obtained from representations of infinitesimal
braids, with the help of Drinfeld associators. We set a dictionary between
representation-theoretic properties of these two structures, and tools to
describe the representations thus obtained. We give an explanation for the
frequent apparition of unitary structures on classical representations. We
introduce new objects -- varieties of braided extensions, infinitesimal
quotients -- which are useful in this setting, and analyse several of their
properties. Finally, we review the most classical representations of the braid
groups, show how they can be obtained by our methods and how this setting
enrich our understanding of them.
