The prolonged attacks occurred on the morning of October 25 last year, on the E3 and 70 bus routes, the police said.
The offences took place between 10am and 11am, and began when the victim, a 24-year-old woman on her way to work, was sat on the lower deck of an E3 bus.
Officers have released CCTV images of a person they wish to speak to in connection to the incident.
A man sat down next to her and began stroking her thigh and grabbing her hands, Metropolitan Police said in its appeal.
She asked him to stop and moved seats, but the man followed and sexually assaulted her.
She again told him to stop but he refused, so she got off the bus outside Acton High School.
The suspect followed her, placing his arms around her and asking for the victim’s phone number.
She boarded the 70 bus towards Acton Lane, but was followed again onto the bus by the suspect, who proceeded to sexually assault her again as she sat on the lower deck.
The victim got off the bus but was still followed, with the man this time grabbing her by the shoulders and attempting to kiss her.
She pushed him and yelled for him to go away, before completing her journey to work, where police were called.
The suspect is described as between 50 and 60 years old, slim, with a small face and pointed nose.
He had greying hair and a distinctive scar or birthmark on his left hand, and was wearing a grey or dark green army hat, dark jeans and a black zip-up jacket.
Anyone who can help, or recognises the person in the photo, should call Detective Constable Natalie Baynes on 07500 607652 or via 101.
Alternatively, call Crimestoppers anonymously on 0800 555 111 or you can tweet @MetCC on Twitter with any information.
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