A partial Latin square L is an nxn array in which n symbols are placed in such a way that no symbol occurs more than once in any row or in any column (not every cell necessarily containing an entry). The question I shall be considering is whether there is an nxn Latin square M on the same set of symbols as L with the property that in no cell do L and M have the same symbol. Note that all cells in M contain a symbol. This question was considered earlier by Chetwynd, Haggkvist and others. It is strongly related to various problems about edge-list colourings, and to work on the Hall index by Cropper and myself. These connections and some other related matters will be brought out in the talk.