Extreme geophysical events such as asteroid impacts and giant landslides can generate mega-tsunamis with wave heights considerably higher than those observed for other forms of tsunamis. In this paper, we review recent advances in the study of mega-tsunamis in the geological record, focusing on well-documented examples that have captured particular attention over the past decade. We provide up-to-date background on the source mechanisms of tsunami generation during asteroid impacts and ocean-island landslides, which are the largest landslides on Earth. We also discuss the main sources of uncertainty for modelling such mega-tsunamis, and for addressing associated hazards.