The well-respected German museum director died on Sunday, August 6 after a serious illness, according to Deutsche Presse-Agentur.
He ran the museum from 2011-2016 and under his leadership the V&A was named 2016 Art Fund Museum of the Year.
Roth had been a supporter of the Remain campaign in the run-up to the EU Referendum and left the museum in September 2016, one year before the official end of his tenure, due to the Brexit vote.
He was the museum's first foreign director and was replaced by former MP Tristram Hunt.
Born in Stuttgart in 1955, Roth held a number of senior roles in museums in Germany during his career including at the Dresden State Art Collections.
V&A chairman Nicholas Coleridge paid tribute to Roth and said the museum was “extremely saddened” by the news.
He said: “Martin made it his mission to raise the international profile of the museum... He was instrumental in the founding of the V&A Research Institute, and was closely involved in its expansion… This, combined with exhibitions such as David Bowie Is and Alexander McQueen: Savage Beauty, Disobedient Objects, raised the V&A to new heights.
“We will greatly miss Martin, and are profoundly grateful for his considerable contribution to the V&A.”
National Portrait Gallery director Nicholas Cullinan said: “I admired Martin Roth’s considerable achievements… He had tremendous energy and vision and will be greatly missed. My thoughts are with his family and friends at this time.”