  In this paper we survey some work on representations of $B_n$ given by the
induced action on a homology module of some space. One of these, called the
Lawrence-Krammer representation, recently came to prominence when it was shown
to be faithful for all $n$. We will outline the methods used, applying them to
a closely related representation for which the proof is slightly easier. The
main tool is the Blanchfield pairing, a sesquilinear pairing between elements
of relative homology. We discuss two other applications of the Blanchfield
pairing, namely a proof that the Burau representation is not faithful for large
$n$, and a homological definition of the Jones polynomial. Finally, we discuss
possible applications to the representation theory of the Hecke algebra, and
ultimately of the symmetric group over fields of non-zero characteristic.
