Mapping between infinite sets

Find a mapping with the following properties.

Variant
A bijection between \(\mathbb N\) and \(\mathbb Z\).
Hint
Try to map the elements of \(\mathbb N\) in an alternating way.
Variant
A bijection between \(\mathbb N\) and \(\mathbb N^2\).
Hint
In other words, your task is to traverse all elements of \(\mathbb N^2\) (omitting none, visiting each element exactly once).
Variant
An injection from \(\mathbb Q\) to \(\mathbb N\). (Or even construct a bijection.)
Hint
Try to use the preceding variant somehow.