The blaze was reported to the fire services at 6:18am on Tuesday morning, with appliances on the scene within minutes.
Images shared on social media show smoke billowing from buildings on both George IV Bridge and Candlemaker Row.
Emergency services have warned the public to avoid George IV Bridge, Candlemaker Row and neighbouring streets as they tackle the blaze.
At least one person has been taken to the Royal Infirmary of Edinburgh.
The extent of the damage caused by the fire remains unclear.
A number of businesses in the vicinity will be closed until further notice and bus services have been diverted.