The irreducible complex representations of degree \(\leq n-1\) of the braid group \(B_n\) on \(n\) strings are classified. For \(n\neq 4,5,6\), they are either one-dimensional, or the tensor product of a one-dimensional representation with a composition factor of a specialization of the Burau representation. The variety of degree three semisimple representations of \(B_4\) is described. It has six components, three of dimension two and three of dimension three.