How many different knight's tours are there? How many solutions does the Knight’s Tour puzzle have when the knight first moves from square a1, through all squares (on each square once), and return back to a1? Please give some solutions. Also, does the Knight’s Tour have any solutions for starting from any random square?
Wikipedia
quotes several sources for a count of 26,534,728,821,064 for the number
of closed directed tours of the 8x8 board.  As Brian Towers notes
in his answer, that is equivalent to the count of Knight's tours
starting from some given square (such as a1) and ending on the same square.
The same Wikipedia page exhibits several such tours.