Quasi-periodic pulsations (QPPs) are a phenomenon commonly observed in solar flares, and are also occasionally observed in stellar flares. They are time variations in the intensity of the flare emission that repeat with approximately constant timescales, or timescales that increase or decrease monotonically in the special case of non-stationary QPPs. There are two main reasons for the interest in QPPs. First is the potential for the diagnosis of plasma parameters in the corona, such as the magnetic field strength and plasma density, which are otherwise difficult to observe directly. If the mechanism causing the QPPs can be inferred, then they would join MHD oscillations of coronal loops as a coronal seismology tool (e.g. Nakariakov & Ofman 2001). Secondly, since QPPs have been found to be a common phenomenon in flares, flares cannot be fully understood without knowing the origin of QPPs. This thesis presents statistical studies of QPPs in both solar and stellar flares, with the aim of learning more about the nature of this phenomenon.

The robust detection of QPPs in solar and stellar flares has been the topic of recent debate. In light of this, this thesis shows how a statistical method described by Vaughan (2005), originally developed to test for the presence of periodic variations of the X-ray emission from Seyfert galaxies, can be adapted to aid with the search for QPPs in are time series data. The method identifies statistically significant periodic signals in power spectra, and properly accounts for red noise as well as the uncertainties associated with the data. The method has been further developed to be used with rebinned power spectra, allowing QPPs whose signal is spread over more than one frequency bin to be detected. An advantage of this method is that there is no need to detrend the data prior to creating the power spectrum. Examples are given where the method has been applied to synthetic data, as well as real flare data from the Nobeyama Radioheliograph (NoRH). These show that, despite the transient nature of QPPs, peaks corresponding to the QPPs can be detected at a significant level in the power spectrum without any processing of the original time series data, providing the background trends are not too steep.

The properties of a set of solar flares originating from a single active region (AR) that exhibit QPPs were investigated. In particular, any indication of QPP periods relating to AR properties was searched for, as might be expected if the characteristic timescale of the pulsations corresponds to a characteristic length scale of the flaring structure. The three AR properties used for this study were the area, bipole separation distance, and average magnetic field strength, which were all measured at the photosphere using SDO/HMI magnetogram data. The AR studied, best known as NOAA 12192, was unusually long-lived and persisted for over three Carrington rotations. During this time a total of 181 flares were observed by GOES. Data from the GOES, SDO/EVE, Fermi, Vernov and NoRH observatories were used to determine if QPPs were present in the flares. For the soft X-ray GOES and EVE data, the time derivative of the signal was used so that any variability in the impulsive phase of the flare was emphasised. Power spectra of the time series data, without any form of detrending, were inspected and flares with a peak above the 95% confidence level in the power spectrum were labelled as having candidate QPPs. The confidence levels were determined taking full account of data uncertainties and the possible presence of red noise. A total of 37 flares (20% of the sample) showed good evidence of having stationary or weakly non-stationary QPPs, and some of the pulsations can be seen in data from multiple instruments and in different wavebands. Because of the conservative detection method used, this may be a lower bound for the true number of flares with QPPs. The fact that a substantial fraction of the flare sample showed evidence of QPPs, using a strict detection method with minimal processing of the data, demonstrates that these QPPs are a real phenomenon that cannot be explained by the presence of red noise or the superposition of multiple unrelated flares. No correlations were found between the QPP periods and the AR area, bipole separation distance, or average magnetic field strength. This lack of correlation with the AR properties implies that the small-scale structure of the AR (which was not accounted for in this study) is important and/or that different QPP mechanisms act in different cases.

Flares that are orders of magnitude larger than the most energetic solar flares have been observed on Sun-like stars, raising the question of whether the same physical processes are responsible for both solar and stellar flares, and hence whether the Sun is capable producing a devastating superflare. A study of QPPs in the decline phase of white-light stellar flares observed by Kepler was embarked upon. Out of the 1439 flares on 216 different stars detected in the short-cadence data using an automated search, 56 flares were found to have QPP-like signatures in the light curve, of which 11 had stable decaying oscillations. No correlation was found between the QPP period and the stellar temperature, radius, rotation period, or surface gravity, suggesting that the QPPs are independent of global stellar parameters. Hence they are likely to be the result of processes occurring in their local environment. There was also no significant correlation between the QPP period and flare energy, while there was evidence that the period scales with the QPP decay time for the Gaussian damping scenario, but not to a significant degree for the exponentially damped case. This same scaling has been observed for MHD oscillations on the Sun, suggesting that they could be the cause of the QPPs in those flares. Scaling laws of the flare energy were also investigated, supporting previous reports of a strong correlation between the flare energy and stellar temperature/radius. Additional analysis was performed on one flare with a rare multi-period QPP pattern. Two periodic signals were identified using the wavelet and autocorrelation techniques. The presence of multiple periods is an indication that the QPPs might have been caused by magnetohydrodynamic oscillations, and suggests that the physical processes operating during stellar flares could be the same as those in solar flares.