At and above the level of measurability, large cardinal notions are typically characterized by the existence of certain elementary embeddings of the universe into an inner model. We may contrast this with smaller large cardinal notions, whose characterizations tend to be strictly combinatorial. In this series of talks, we survey results from Magidor's thesis, in which he shows that the large notion of supercompactness can also be viewed combinatorially, and in this light supercompactness is seen to be a natural strengthening of ineffability. We will also survey modern results which show how these strong combinatorial principles can be forced to hold at small successor cardinals.