Melbourne University or Unimelb is currently rated the best uni in Australia and roughly 37th in the world.
It was first established in 1853 at the height of the gold rush and opened its doors in 1855 with 4 lecturers and 16 students, of whom only 4 graduated.
The population of Melbourne at the time was about 40 000.
Currently the Uni has about 38 000 students at over a dozen places.
For more detailed info, see Wikipedia.