The history of potential vorticity developments is reviewed and the invertibility principle for potential vorticity is presented. We discuss the various versions of potential vorticity and the Lagrangian conservation properties in the differently approximate models. Some new properties of potential vorticity are summarized. Several methods of potential vorticity inversion are discussed and compared. A lot of examples are examined and their disadvantage are also investigated. We conclude the remark by predicting some possible applications of the IPV concept and potential vorticity inversion.