Both of the attacks took place in the Leith area of the city in the early hours of yesterday.
The first incident happened shortly after midnight on Easter Road when a 30-year-old woman was putting a rubbish bag out onto the street.
She was approached by the suspect, who then grabbed her and followed her back to her flat. He ran off after the victim raised the alarm and headed onto the cycle path behind Leith Academy.
The second incident happened around 1am on Henderson Street when a 23- year-old woman was getting off a bus.
As she walked towards Giles Street the suspect approached her, grabbed her with both arms and indecently assaulted her.
She managed to scream for help and he was disturbed by a witness in a neighbouring property.
The suspect is described as white, 6ft tall, slim to medium build, 20 to 30 years old, short blond hair, wearing a dark grey coloured jacket, possibly with a hood, dark jeans and a German or Eastern European accent.
Detective Inspector Morag Bruce of Police Scotland said: "Incidents like these are extremely rare, but the victims were left shaken and distressed.
"If anyone has any information or recognizes the description of the suspect, they are urged to contact police immediately."
Police are appealing for anyone with information that can assist them with their inquiries to contact Police Scotland on 101, or Crimestoppers in confidence on 0800 555 111.