3x3 corner in 4 moves (each square visited excactly once):

 3  .  .  2
 0  .  .
 .  .  4
 1

8x8 chessboard in 14 moves (each square visited at least once):

13  5  8  .  .  7 10  2
 .  .  .  .  .  .  .  .
 .  .  .  .  .  .  .  .
 .  .  .  .  .  .  .  .
 .  6  .  .  .  .  9  .
 .  .  3  .  .  .  4  .
11  .  .  .  .  . 12  .
 0  .  .  .  .  .  .  1    with 0=14

 solution of Sam Loyd 1876 (not unique)