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Over the last century statistical seismology has been one of the key methods to describe the nature of the seismic process. The clustering of earthquakes in space and time has been established and is included in statistical models such as the Epidemic Type Aftershock Sequences (ETAS) model to describe earthquake occurrence. On the other hand the clustering in magnitude is considered to be zero. If magnitude correlation exist and included in the statistical models this could lead to better interpretation of the seismicity in the region that can be used to update and improve probabilistic forecasts of seismicity. This research seeks to test the existence or not of earthquake magnitude correlations by analysing earthquake catalog data from different regions worldwide. In standard models of seismic hazard such as ETAS, the magnitude of each earthquake is independently drawn from a Gutenberg-Richter distribution. As such, there should not be any correlations between magnitude and seismicity if the usual ETAS assumptions are correct, since the increase in the seismicity only affects the number of aftershocks, while having no effect on their magnitude. We have shown that the mean magnitude of earthquakes is larger during periods of high seismicity confirming the existence of correlations. I additionally studied acoustic emission (AE) data from laboratory triaxial rock deformation experiments. The AE technique provides a means to analyse microcracking activity inside the rock volume and since experiments can be performed to simulate crustal conditions, AE can be used as a proxy for natural processes such as earthquakes. The results have also been analysed in terms of crack length and stress intensity. Finally, two statistical models, the ETAS model which is most widely used in the field of statistical seismology and a compound poisson model have been tested on their forecasting capabilities.