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This year is 400 years since Shakespeare's death in 1616. I want to visit, for my first time, London in the autumn (approximately in September).
Can someone recommend some good performances and events (or probably some festivals) in London regarding to Shakespeare's works and links to the events?
PS I have found that in September it is possible to see King Lear by the Royal Shakespeare Company.
'Recommendations' are off-topic for here as they're more opinion based. But for just finding performances and exhibitions related to the anniversary year there's this site: shakespeare400.org. It looks like you've already found the Royal Shakespeare Company website.
Shakespere is performed throughout the UK. You might be interested in some outdoor performances (msotly July/August, but some are probably still happenng in September). Also, a journey to Stratford-upon-avon is probably a must for any shakespeare fan.