A family representative said he “passed away peacefully at home from natural causes”.
Rogers topped pop and country charts during the 1970s and 1980s, and won three Grammy awards.
Known for his husky voice and ballads including The Gambler, Lucille and Coward Of The County, his career spanned more than six decades.
He once summed up his popularity by explaining that he believed his songs “say what every man wants to say and that every woman wants to hear”.
After growing up in poverty on a federal housing estate in Houston, Texas, Rogers began recording with a string of bands, including Kenny Rogers and the First Edition, before launching his solo career in 1976.
He was never a favourite of music critics, but became one of the most successful pop-country crossover acts of all time, and the 10th best-selling male artist in US history in terms of album sales.
He collaborated with other country music legends during his career, including Dolly Parton and Willie Nelson.
In 2007 he unexpectedly found himself back in the limelight in the UK when The Gambler became the unofficial World Cup anthem of England’s Rugby Team.
The song became so popular that during his 2013 Glastonbury Festival legends slot Rogers played it twice.
That same year, he was inducted into the Country Music Hall of Fame and received a Lifetime Achievement Award from the Country Music Association.
In their statement his family said he had “left an indelible mark on the history of American music”.
A keen businessman, Rogers led several ventures over the years, mainly in property and the restaurant sector.
He also acted in several movies and TV shows, including starring as a race car driver in the 1982 movie Six Pack.
During an interview with the BBC in 2013, he recalled his “obsession” with tennis, and said he became so good that he ranked higher than Bjorn Borg in the ATP’s doubles table.
He was married five times and had five children.