The London theatre scene is amazing. Sign up to mailing lists to stay as up to date as possible. The below are some good places to start.
Barbican
Address: Barbican, Silk Street, London EC2Y 8DS
Telephone: 020 7638 4141
Website: http://www.barbican.org.uk/
Description: Post war complex in Clerkenwell that houses art exhibits, concerts and conferences.Donmar Warehouse
Address: 41 Earlham St
Telephone: 0 844 871 7624
Website: http://www.donmarwarehouse.com
Description: Tiny, brilliant theatre in Seven Dials, with standing tickets at £7.50 and £10 front row tickets released at 10AM every Monday.Globe Theatre
Address: 21 New Globe Walk, Bankside, London SE1 9DT
Telephone: 020 7902 1400
Website: http://www.shakespearesglobe.com
Description: Replica of the original Shakespearian Globe Theatre. £5 to stand and see a play as a groundling.Lastminute.com
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Website: http://www.lastminute.com
Description: Good website to find cheap deals for theatre/musical shows.National Theatre
Address: National Theatre, Upper Ground, South Bank, London SE1 9PX
Telephone: 020 7452 3000
Website: http://www.nationaltheatre.org.uk
Description: One of the brutalist structures on the South Bank of London, the NT puts on world class productions. If you are under 26, sign up to the Entry Pass program to have access to amazing tickets for £5.Southbank Centre
Address: Southbank Centre, Belvedere Rd, London SE1 8XX
Telephone: 0844 875 0073
Website: http://www.southbankcentre.co.uk/
Description: A staple of the South Bank. Has a constantly changing line-up of festivals and markets. Good café spot for studying.Time Out
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Website: http://www.timeout.com/london
Description: Legendary website chock full of London recommendations. Also has ticket deals on plays and restaurants that are worth checking. Sign up to the mailing list to stay up to date.