The Glasgow-born journalist and television presenter, who presents BBC One's Sunday politics talk show The Andrew Marr Show and travelled the world for his series Andrew Marr's History of the World, took ill on Tuesday.
The BBC said in a statement: "Andrew Marr was taken ill yesterday and taken to hospital. The hospital confirmed he has had a stroke.
"His doctors say he is responding to treatment. His family have asked for their privacy to be respected as he recovers.
"We will continue to broadcast The Andrew Marr Show and Radio 4's Start The Week with guest presenters in his absence.
"His colleagues and the whole BBC wish him a speedy recovery."
Politicians also wished Marr a quick recovery.
His illness came just two days after he interviewed Prime Minister David Cameron about the Coalition Government's mid-term report on his show.
Labour Ed Milliband said: "My thoughts are with Andrew and his family. I hope he gets well soon."
Scottish BBC presenter Andrew Neil said: "Very distressed to hear the news about Andrew Marr. Best wishes for a full and speedy recovery."
Sky TV's political editor Adam Boulton said: "Best wishes for a speedy recovery."
Strokes come in several different forms, and are caused by a blockage to the brain's blood supply. Many sufferers make a full recovery, but strokes can lead to a range of debilitating conditions such as loss of speech, impaired movement, memory loss or paralysis.
It is not known how severe Mr Marr's condition is, but it is likely he will need an intensive period of therapy and a long period of convalescence.
The Stroke Association said it was deeply saddened to learn the BBC presenter had suffered a stroke.
The charity said one-quarter of the 150,000 people who are affected by the condition each year in the UK were of working age.
Joe Korner, director of communications at the Stroke Association, said: "We are deeply saddened to hear about Andrew Marr's stroke and our thoughts are with him and his family at this hard time.
"A stroke happens in an instant but the effects can often last a lifetime.
"However, with the right care and support it is possible to make a recovery and return to a life after stroke."
Marr, 53, is the corporation's former political editor. He began his career as a newspaper reporter, becoming the editor of The Independent before turning to television.
He has also presented a number of history programmes along with his politics show and has had five books published.
He is married to fellow journalist Jackie Ashley, and the couple have three children.
In May 2011 it emerged he had taken out a super-injunction to prevent the reporting of an alleged affair he had with a political journalist.