For official, rated events, yes.
Touraments

Primarily it is to assist with resolving possible disputes and verifying draw claims.
I should have also clarified, in my initial reply, that blitz doesn't require notation.
Ah. So if I've wrote down the game I can claim a draw under the three fold repatition rule if it's happened or the 50 move rule?

I don't know how FIDE handles draw claims and incomplete scoresheets but for the USCF you have to have one to claim the draw.

It is definitely to your own advantage to accurately record all the moves of both players. If this is new to you, practice notating ahead of time. If your scoresheet is incomplete, noticeably inaccurate, or hard to read, the tournament director/arbiter will give priority to your opponent's scoresheet (which may itself be innacurate and work against you). Just like in court, defend your rights by having accurate and complete information.
Besides, don't you want a record of the game so that you can enjoy or analyze it later?
So every time I go down my local chess club I have every right to récord the game? Will do it in the future.
Hi - is it compulsary to write down your chess moves if you play in tournaments?