The Laws of Cricket say in Law 5:
In a match of more than one day’s duration, the captain of the fielding side may demand a new ball after the prescribed number of overs has been bowled with the old one. The Governing Body for cricket in the country concerned shall decide the number of overs applicable in that country, which shall not be less than 75 overs.
So in Twenty20 and one-day matches, there will generally not be a new ball in the Test Match sense.
There is the rule that in ODIs, two new balls are used, one from each end, but they are never replaced by a new ball (unless damage occurs). The previous rule was that the one ball is replaced after 34 overs, but with another used ball, not a new one, and this replacement was manadatory, not at the request of the fielding side, as in the law.
In Test Matches, the law5 applies, and a new ball is available after 80 overs, per ICC playing conditions. In English first-class cricket, the new ball is also available after 80 overs. I haven't looked up all the other domestic first-class competitions, but the numbers are probably similar.
You are not required to take the new ball. You can use the old one as long as you like.