Transport algorithms of numerical ocean circulation models are frequently exhibiting truncation errors leading to spurious diapycnal mixing of water masses. The chapter discusses methods that might be useful in diagnosing spurious diapycnal mixing and describes some approaches that might be helpful for its reduction. The first one is related to the use of the Arbitrary Lagrangian Eulerian (ALE) vertical coordinate which allows the implementation of vertically moving meshes that may partly follow the isopycnals even if the basic vertical coordinate differs from isopycnal. The second approach relies on modified advection schemes with the dissipative part of the transport operators directed isopycnally. Finally the third approach deals with new efficient and stable advection algorithms of arbitrary high order based on the WENO- ADER method, which can be applied to both structured and unstructured meshes. While practical benefits of using the reviewed approaches depend on applications, there are indications that equipping present state- of-the-art ocean circulation models with them would lead to reduced spurious transformations.