Scientists at the University of Leicester confirmed the news at a press conference in the city.
The university's lead archaeologist Richard Buckley said the tests proved the remains were the king's "beyond reasonable doubt".
Deputy registrar Richard Taylor described the discovery as "truly astonishing".
Archaeologists previously said there was strong circumstantial evidence to suggest the bones, exhumed from a car park behind social services offices in the city are those of the Plantagenet king but did not want to make any academic decision before the skeleton was subjected to a number of tests.
The skeleton, with a metal arrow in its back and severe trauma to the skull, was exhumed in September last year during an archaeological dig.
It was found in the same area of what was Grey Friars church where Richard III was recorded to have been buried after his death at the Battle of Bosworth in 1485, the last major act in the Wars of the Roses.
Initial examinations showed the bones to be those of an adult male and the remains were said to be in a good condition.
The skeleton had a curved spine, consistent with accounts of Richard III's appearance.
DNA taken from the skeleton has been analysed and compared with that of Michael Ibsen, a descendant of Richard III's family. Radiocarbon tests and genealogical studies have also taken place.
Richard III's demise was dramatised by Shakespeare, who had the king calling out "a horse, a horse, my kingdom for a horse" before he was killed on the battlefield.
There were cheers from media who had gathered from around the world as the announcement was made at the University of Leicester this morning.
University archaeologist Dr Jo Appleby said the skeleton was found in good condition with its feet missing in a grave.
Its hand were crossed over the front of the pelvis and there was no evidence of a coffin or shroud found with the skeleton.
Dr Appleby, an osteoarchaeologist based at the University's School of Archaeology and Ancient History, said: "Taken as a whole the skeletal evidence provides a highly convincing case for identification as Richard III.
"The analysis of the skeleton proved that it was an adult male but was an unusually slender, almost feminine, build for a man.
"This is in keeping with historical sources which describe Richard as being of very slender build.
"There is however no indication that he had a withered arm - both arms were of a similar size and both were used normally during life."
Richard III was sometimes depicted historically with a withered arm and hunched back.
"The skeleton is that of an individual aged between the late twenties and late thirties. We know that Richard III was 32 when he died and this is entirely consistent with the Grey Friars skeleton."
Announcing the results to the media, Mr Buckley said: "It is the academic conclusion of the University of Leicester that the individual exhumed at Grey Friars in August 2012 is indeed King Richard III, the last Plantagenet King of England.
"It has been an honour and privilege for all of us to be at the centre of an academic project that has had such phenomenal global interest and mass public appeal. Rarely have the conclusions of academic research been so eagerly awaited."
University of Leicester geneticist Dr Turi King confirmed DNA from the skeleton matched that of two of Richard III's family descendants - Canadian born furniture maker Michael Ibsen and a second person who has asked to remain anonymous.
Dr King said: "The DNA sequence obtained from the Grey Friars skeletal remains was compared with the two maternal line relatives of Richard III.
"We were very excited to find that there is a DNA match between the maternal DNA from the family of Richard III and the skeletal remains we found at the Grey Friars dig."
Analysis of the skeleton showed that the individual was male and in his late 20s to late 30s. Richard III was 32 when he was killed at the Battle of Bosworth in 1485.
The analysis also showed the individual had a slender physique and severe scoliosis - a curvature of the spine - possibly with one shoulder visibly higher than the other. This is consistent with descriptions of Richard III's appearance from the time, the researchers said today.
Trauma to the skeleton showed the king died after one of two significant wounds to the back of the skull - possibly caused by a sword and a halberd.
Dr Appleby said this was consistent with contemporary accounts of the monarch being killed after receiving a blow to the head.
The skeleton also showed a number of non-fatal injuries to the head and rib and to the pelvis, which is believed to have been caused by a wound through the right buttock.
Dr Appleby said these may have been so-called "humiliation injuries" inflicted after his death.
"The skeleton has a number of unusual features: its slender build, the scoliosis and the battle related trauma," she said.
"All of these are highly consistent with the information that we have about Richard III in life and about the circumstances of his death.
"Taken as a whole, the skeletal evidence provides a highly convincing case for identification as Richard III."
Hobbit star Richard Armitage is now set to star as the King in a film about his life. The script has been written by Scottish researcher Philippa Langley, who led archaeologists to the spot where his skeleton was found.