Given an ordinal (or cardinal) $x$:
$x + 0 = 0 + x = x$.
$x \cdot 0 = 0 \cdot x = 0$
$x^0 = 1$
$0^x = 0$ (if $x > 0$)
Some set theorists (e.g. Thomas Jech) instead classify zero as a limit ordinal, because like other limit ordinals, it is the limit of all ordinals less than it (of which there are none).