Foley, who played 62 times for his country, was found dead in the team's hotel in Paris, ahead of a match in the European Champions Cup against Racing 92.
A statement by the Irish Rugby Football Union said: "The IRFU and Munster Rugby pass on our deepest sympathies to all of Anthony's family and friends and ask for privacy for the family at this sad time." No cause of death has been given.
Foley had been on the backroom staff with Munster since 2009, having made over 200 appearances as a player for the provincial side, skippering them to victory in the 2006 Heineken Cup, Europe's top club competition.
Sunday's match in Paris was immediately postponed as the news of his passing became public, with a wreath and Munster scarf laid on the center circle of the stadium where the team had been due to play.
Tributes were led by Irish president Michael Higgins. "It is with great sadness that I heard of the sudden death of Anthony Foley, the Munster rugby team's head coach and one of the great figures of Irish sport in the modern era," he said.
"He was regarded with great respect and deep affection not just among the Munster rugby fans but by all those interested in Irish sports and those with whom they interacted abroad," added Higgins, who is also patron of the Irish Rugby Union.
Back-row forward Foley scored a try on his debut against England in 1995 and became a key figure in the Irish side, captaining them on three occasions.
England World Cup winner Brian Moore, who played against Foley, said: "Just heard the terrible news of the death of Munster's Anthony Foley -- thoughts with his family -- very sad indeed."
Foley's father Brendan, who was with the Munster party in Paris at the time of his son's death, won 11 caps for Ireland and was in the Munster team which famously beat the All Blacks in 1978.
Foley's sister Rosie was also member of the Irish women's squad
He leaves a wife Olive and two sons.
Clubs across Britain and Europe added sympathy messages on social media, while Ireland's former track and field star Sonia O'Sullivan said his death was "sad and shocking".
Stunned Munster fans in Paris gathered for their own poignant tribute with a moving rendition of the Fields of Athenry.
An impromptu memorial with messages of condolence was also being left outside the match stadium.