In this project, pulsar J0034-0721 was observed at 1.4 GHz using the Parkes telescope in October 2014. This pulsar is known to show the subpulse drifting phenomenon where the subpulses shift in rotational phase such that each rotation shows a pulse with a different shape. Two different patterns of subpulse drifting were detected at this frequency (mode A and B). Detailed polarisation analysis was performed on these two drift modes. For each mode, the drifting subpulses and other non-periodic emission were compared in both total and linear polarisation intensity of the pulsar emission. It was found that in mode A the relative strengths of drifting subpulses and non-periodic emission are similar in total intensity and linear polarisation while in mode B, the strength of drifting subpulses compared with non-period emission was found to be stronger in total intensity but weaker in linear polarisation. The emission of both mode A and mode B was found to be depolarised and the depolarisation was proven to be caused by more than just the superposition of two purely linearly polarised orthogonal polarisation modes (OPMs). For the first time for this pulsar, periodic transitions have been found between the two OPMs related to the subpulse drifting periodicity. The values of the linear polarisation position angle (PA) of the two OPMs were found to be separated by about 85 degree rather than being strictly orthogonal. The PA against rotational phase curves were found to be different in the two drift modes. Although this can be partially explained by different dominating OPMs, it is not enough to fully explain the observed differences. Also, relevant theories have been reviewed and applied to explain the observational results of PSR J0034-0721.