In this thesis the closely related problems of embedding symmetric latin rectangles and embedding properly edge-coloured complete graphs are considered. The thesis is divided into three parts. The first part is an introductio~ and survey about completing partial latin squares and embedding latin rectangles. The second part is a proof of a former conjecture of Dugdale and Hilton about embedding edge-coloured complete graphs and the third and final part is a partial generalisation of the second part to the problem of embedding symmetric latin squares.