A term rewrite system (TRS) terminates if, and only if, its rules are contained in a reduction ordering $>$. In order to deal with {\em any\/} set of equations, including inherently non-terminating ones (like commutativity), TRS have been generalized to {\em ordered\/} TRS $(E,>)$, where equations of $E$ are applied in whatever direction agrees with $>$. The confluence of terminating TRS is well-known to be decidable, but for ordered TRS the decidability of confluence has been open. Here we show that the confluence of ordered TRS is decidable if $>$ belongs to a large class of path orderings (including most practical orderings like LPO, MPO, RPO (with status), KNS and RDO), since then {\em ordering constraints\/} for $>$ can be solved in an adequate way. For ordered TRS $(E,>)$ where $E$ consists of {\em constrained equations\/}, confluence is shown to be undecidable. Finally, also {\em ground reducibility\/} is proved undecidable for ordered TRS.