Sosanya has appeared in many roles in the theatre, on television and in films. Her first big break in theatre was in Anthony and Cleopatra at the National Theatre, and with the 2001 series Teachers. She also appeared in Sorted, People Like Us, Love Actually, Nathan Barley, Renaissance, Casanova, as Karen Blaine in the Jonathan Creek episode "The Three Gamblers", Much Ado About Nothing, Cape Wrath/Meadowlands, the Doctor Who episode "Fear Her", and FM.
In 2003, she played Rosalind in As You Like It with the Royal Shakespeare Company (RSC) at the Swan Theatre, Stratford-upon-Avon.In 2008, she returned to the RSC to play Rosaline in Love's Labour's Lost and, in 2009, appeared in a radio adaptation of a story from the short story collection The State of the Art.
She reads the part of Sephy on the audio book versions of Malorie Blackman's Noughts and Crosses series.
In January 2010, Sosanya appeared as Mae Pollock in Tennessee Williams' play Cat on a Hot Tin Roof at the Novello Theatre, London.
She appeared as Colly Trent in series 2 of the BBC television drama Five Days. She appeared in the BBC4 television series Twenty Twelve, a comedy about the London 2012 Olympic buildup, and the BBC1 drama series Silk and Hustle. She also made a brief appearance in the children's CBBC science fiction series, Wizards vs Aliens as Benny Sherwood's mother, Trisha.
Sosanya played the character Alibe Silver in Treasure Island (2012). She played a main character, Kate McKenzie, in series one to three of the BBC original drama Last Tango in Halifax during 2012–2015,and a main character, Lucy Freeman, in the TV series W1A in 2014. In 2015, she starred with Catherine Tate and Mark Gatiss in a new play, The Vote, in the run-up to the UK 2015 general election. In 2016, she starred in the play Young Chekhov trilogy and as DCI Laura Porter in the ITV series Marcella.
In 2014, she appeared in the world premiere production of Privacy at the Donmar Warehouse, London. In 2016, she appeared in the world premiere production of Elegy at the Donmar Warehouse. She had recurring appearances in episodes of Shetland.
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