The first attack took place at 3:45am on Sunday on Queensferry Street, when a 21-year-old woman and her 19-year-old male friend were waiting for a taxi outside a Greggs bakery.
A large group of teenage boys and girls suddenly launched an assault on them.
Both victims then made their way along Queensferry Road where the man was again punched and kicked while some of the suspects struggled with the woman.
A passing motorist assisted both victims and the gang ran off westwards on Queensferry Road.
The woman's personal belongings in her handbag were stolen and the teenage victim was treated at the Edinburgh Royal Infirmary for facial injuries.
Earlier that night, a 24-year-old man was assaulted by about four teenage boys in the same area.
The suspects spoke to the man before punching him in the face. He was taken to the ERI to be treated for a suspected fractured cheekbone and broken nose.
Police believe the same individuals may be responsible for both incidents.