My dictionary says the word downtown is usually used in America.

What words are used in Britain?

A: Where are you going?

B: I'm going to ~~
I've heard "city centre" used in the UK to mean the oldest central district (actually it's called the City of London, or simply "The City".)
Hi, AS.

Ah... so we can simply say 'I'm going to the city?'
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In London, yes. "The City" is a specific geographic area with a proper name.

In the US, downtown and uptown are generic names for the commercial center of the city and the near-central residential area of a city.
You can say "central London/Amsterdam/Paris" etc.
Ah// then if we say 'I am going to central London/Amsterdam/Paris,' this may sound someone not from those cities will go there. So, that is not likely to be heard from the locals, is it?

I think I heard some Australians saying 'I'm going to the city' or something like that.
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If you are in London, you can say "I'm going to the West End"

That is where the shops, theatres, cinemas and clubs are, in the centre of the town

In London, "the City" means something slightly different - it is where the big business, banking and finance is

If you said "He has an important job in the City" it might mean "He is a well-paid accountant". But it probably wouldn't mean "He runs a popular music club"

Best regards, Dave