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President of Ireland, Michael D Higgins, has sent his condolences to Sir David's family.
He said he was killed while carrying out a fundamental role of a politician - helping their constituents.
He offered his "deepest sympathies to the family, friends, colleagues in Parliament and constituents" of Sir David.
He also extended thoughts to the family of the late Jo Cox MP, murdered just over five years ago in her constituency of Batley and Spen.
"On behalf of the Irish people, I send them our deepest sympathies."
The British-Irish Parliamentary Assembly (BIPA) has cancelled a scheduled plenary meeting at Westminster on Monday as a mark of respect for Sir David.
By Alex Taylor
BBC Entertainment reporter
Reaction to the death of Sir David Amess has been coming in from around the world.
Overnight, Australia's Home Affairs Minister Karen Andrews tweeted: "I am shocked and saddened by the murder of British MP Sir David Amess as he met with his constituents in the UK.
"The Australian Government sends our condolences and sympathies to his family, friends, Parliamentary colleagues and the community he served."
By Alex de Waal
By Lucy Williamson
Ros Atkins examines why tensions between China and Taiwan have increased over recent weeks.
Rising temperatures and desertification are forcing many Mauritanians to leave their ancestral homes in search of a better life.
By Bernd Debusmann Jr
BBC News, Washington
By Enda McClafferty
BBC News NI political editor