The suit alleges the police officers violated Mr Floyd’s rights when they detained him, and that the city allowed a culture of excessive force and racism to take root in its police force.
Mr Floyd’s death is part of a “public health crisis” in black America, said the family’s attorney Ben Crump.
New body camera videos have emerged showing Mr Floyd’s final moments.
In the footage, Mr Floyd is seen pleading with officers as they try to force him into their car, US media say. Mr Floyd reportedly tells police he could not breathe, and asks if he could lie on the ground instead.
A district judge allowed news outlets to view the two videos, which show the clearest picture yet of Mr Floyd’s last moments in the Minnesota city on 25 May.
All of the officers involved have since been fired and charged over his death. Derek Chauvin, who in a separate video filmed by eyewitnesses was seen kneeling on Mr Floyd’s neck for nearly eight minutes, has been charged with second-degree murder. The three other officers – Thomas Lane, J Alexander Kueng and Tou Thao – have been charged with aiding and abetting murder. A Minneapolis judge has set a tentative trial date for the four on 8 March, 2021.
The death of Mr Floyd, a 46-year-old unarmed African American, sparked huge protests across the US against police brutality and racism.
Civil rights attorney Mr Crump announced the Floyd family’s lawsuit against Minneapolis on Wednesday, filed in US District Court in Minnesota.
Mr Crump said the legal action will seek unspecified monetary damages on behalf of Mr Floyd’s children and siblings.
“This complaint shows what we have said all along, that Mr Floyd died because the weight of the entire Minneapolis Police Department was on his neck,” he said in a statement.