Three graph colouring problems are studied in this thesis with the main focus on'acyclic edge colouring problem'. The first part of this thesis deals with some classes of graphs and improved upper bounds are obtained. The second part deals with k-intersection edge colouring. It aims to find the minimum number of colours that are sufficient to colour the edges such that for any pair of adjacent vertices, the number of common colours received on the edges incident on them is at most k.