It happened at Ardmonagh Parade, in Turf Lodge, at around 10.30 PMon Thursday night.
The man was taken to hospital for treatment, however his injuries are not thought to be life threatening.
Detective Inspector Harvey said: “Shortly before 10:30 pm last night it was reported that a 20 year old man was attacked by three men and shot three times in the legs and ankle.
“The victim was then taken to hospital where he received treatment for the injuries sustained during this violent crime.
“This was a brutal and despicable attack, which has left the victim with potentially life changing injuries.
“It is yet another example of how criminal groups seek to control communities through fear and violence.
“I would appeal to anyone who may have noticed any suspicious activity in the area, to contact Detectives at Musgrave Criminal Investigation Branch on the non-emergency number 101, quoting reference 1360 12/04/18.
“Alternatively, information can also be provided to the independent charity Crimestoppers on 0800 555 111 which is 100% anonymous and gives people the power to speak up and stop crime.”
SDLP Councillor Tim Attwood said the “appalling” shooting had the hallmarks of a so-called punishment attack by paramilitaries to control communities.
“In recent days there’s been some sinister graffiti threatening further shootings as well,” he told the BBC’s Good Morning Ulster programme.
“There are some people out there who are intent to take the law into their own hands.”
Mr Attwood added: “There may be some people who were involved in paramilitary activity in the past who still think they can control our communities. Those days are gone.
“The message needs to be loud and clear that there is only one police service…and we can’t have violent or gun attacks like this on our streets anymore.”