An order-$n$ magic square is an $n \times n$ matrix over the numbers $\{1, ... ,n^2\}$, each appearing exactly once, whose row and column sums are all equal. Sometimes the sums of the diagonals are required to be equal too.

These objects have a rich history, and they are hugely popular in recreational math. I remember being fascinated by constructions of magic squares when I was 10. It seems natural to ask how many order-$n$ magic squares are there? An exact formula is probably too much to hope for, but it is probably possible to give some asymptotic bounds.

Has anybody asked this question before? Are there known bounds on the number of order-$n$ magic squares?