Although earthquakes are phenomena of great complexity, some simple general laws govern the statistics of their occurrence. Interestingly, some of these most important laws exhibit scale invariance properties, as the Gutenberg-Richter law, the Omori law and the space clustering of epicentres. If the space and temporal clustering are considered a general and distinct feature of seismic occurrence, the question of the existence of correlations between magnitudes of subsequent earthquakes is intensively debated, since the standard seismological approach assumes independence of earthquake magnitudes. Our recent analysis of the California Catalogue has shown the existence of non-zero spatio-temporal magnitude correlations. Namely, earthquakes tend to occur, with higher probability, close in time, space and magnitude to previous events. Moreover, considering an earthquake as a point event in time, a branching model based on a dynamical scaling hypothesis, relating magnitude to time and space, has been proposed. Our model reproduces the observed magnitude correlations and a number of statistical properties of seismic occurrence, suggesting that these naturally originate from this scaling hypothesis.