It's well-known that a knight placed on one square of a chessboard can get to any other square, but a bishop can only reach half the squares from a fixed starting point. Another question on this site dealt with a new type of chess piece with a different way of moving. I'm trying to generalise all of these ideas.
A knight's move takes you between opposite corners of a $2\times3$ rectangle of squares. A bishop's move takes you between opposite corners of a $2\times2$ rectangle. So let's define a General (a new, generalised breed of chess piece) to be a piece that can move from its current position to the opposite corner of an $m\times n$ rectangle. For what values of $m$ and $n$ is it possible for a General to start from one square and reach every other square on the chessboard?
(Possible further generalisations include changing the chessboard to $M\times N$ rather than $8\times8$; looking for a 'General's tour', the General visiting each square exactly once; or looking for a 'General's tour' with the General visiting each square exactly once and ending on the same square it began on.)