The website of Columbia University is very large and I couldn't find a page in it anywhere outlining it's history; so here goes.
The charter of King's College of 1754, named ministers of five different denominations to it's governing board. The first building was raised near the southern end of Manhattan in 1756. During the Revolutionary War, classes were suspended and the library was stored in the city hall.
In 1767 a school of medicine, the College of Physicians and Surgeons, was founded.
In 1784 the name was changed to Columbia College.
In 1857 the college was moved to Madison Avenue between 49th and 50th Streets.
In 1858 a school of law was opened.
In 1896, Columbia became a university, and one year later, moved to it's present location on Morningside Heights.