Abstract: If $R$ is a field then the conjugacy class of $x\in M_n(R) = End(V)$ is determined by its rational canonical form using the theory of modules over the PID $R[T]$. If $R$ is a discrete valuation ring then the situation is more complicated, even if the characteristic polynomial of $x\in M_n(R)$ is irreducible over the quotient field $K$ of $R$. We discuss the following questions. (1) What further assumptions on $x$ and $R$ are useful? (e.g. $x$ semisimple, $R$ Henselian) (2) How do we sort out non-conjugate elements of $M_n(R)$ that become conjugate in $M_n(K)$? (3) Are some conjugacy classes of $M_n(R)$ better than others? (4) To what extent can $x$ be measured against a canonical form?