Police are appealing for witnesses following the hate crime, which took place yesterday afternoon.
Officers said that, at around 5.10pm, a man was heard making several offensive remarks, including racial abuse, towards passing members of the public as he stood outside the H&M store between Frederick Street and Hanover Street.
They are now appealing for anyone who witnessed this incident to come forward.
Sergeant Grant McCulloch, of the Divisional Violence Reduction Unit, which is based in Leith police station, said: “It is completely unacceptable for innocent members of the public to feel targeted or unsafe.
“From our inquiries it would appear that this individual was not provoked yet chose to shout offensive comments at a number of people including children.
“I want to hear from anyone who remembers this man and in particular a young family that were subjected to racial abuse.
“Tackling hate crime is a priority for Police Scotland and I would urge anyone who has been a victim or witness of this to speak to a local officer, contact Police Scotland on 101 or make an anonymous report through Crimestoppers on 0800 555 111.”