US News rankings
The US News rankings are the most widely used set of college and graduate professional school rankings.
As an undergraduate institution, Columbia typically placed around 8th or 9th, at least until the 2011 rankings were released. This is probably fairly accurate, but many people believed that Columbia was consistently under-ranked because its admissions committee places less of an emphasis on SAT scores than most of the other top 10 universities. Additionally, alumni giving is consistently lower at Columbia than at comparable institutions, further affecting our rank. Somehow, Penn managed to consistently achieve higher rank, which, of course, was a perversion of general logic.
The 2011 rankings placed Columbia in a surprising but not necessarily unwarranted fourth place. This put Columbia in its generally-assumed position relative to the rest of the Ivy League, bested only by Harvard, Yale, and Princeton, but many were genuinely surprised to find Columbia beating even high achieving Stanford and MIT.