Financial indices are located all over the world, including major international cities such as Hong Kong, London, New York, Frankfurt and Tokyo. The range of indices means that no matter where you are located, there will be one open and trading.
So if you live in the UK, for example, your local major index, the FTSE 100 is open from 08:00 to 16:30, by which point indices in New York would still be trading, followed by those in China and Japan at night until the early hours of the morning.
Different indices can operate with different trading hours. As mentioned, the FTSE100 trades from 08:00 to 16:30, but major indices in the USA open from 09:30 to 16:00. The Hang Seng and Nikkei trading hours are divided into two sessions, with an hour’s break for lunch in-between.