  We study centralizers of elements in domains. We generalize a result of the
author and Small, showing that if $A$ is a finitely generated noetherian domain
and $a\in A$ is not algebraic over the extended centre of $A$, then the
centralizer of $a$ has Gelfand-Kirillov dimension at most one less than the
Gelfand-Kirillov dimension of $A$. In the case that $A$ is a finitely generated
noetherian domain of GK dimension 3 over the complex numbers, we show that the
centralizer of an element a $A$ that is not algebraic over the extended centre
of $A$ satisfies a polynomial identity.
