The Completability of partial Latin squares

Abstract: 

As early as 1000 A.D., the arrangement known as the Latin square has been used in artistic cultural expression. Recent uses of these arrays include experimental design, cryptography, optical network designs and, more famously, recreational mathematics and Sudoku. The average Sudoku solver must ask the question `How can I complete this unlled Latin square?'. We explore the more primary and general questions of `Can I complete an unlled Latin square?' and `When can I be sure of this?'. We survey the ways combinatorialists have answered these questions from the classical results of the mid-late 20th Century to the present day and give an outline of methods used in this research.  In particular, we focus on results from Hall, Ryser and Evans and their more recent extensions by Haggkvist and others. Additionally, an outline of some recent applications of completability and approximation theorems is given.