The collision took place around 9.15am this morning (Sunday, May 14) at the junction of Quality Street and Main Street.
A 70-year-old woman was struck by a 41 service Lothian Bus and was pronounced dead at the scene.
Road closures remain in place and motorists are advised to avoid the area while investigations continue at the scene.
Police are now appealing for anyone who saw the accident to get in touch.
Inspector Stuart Mitchell of Drylaw Police Station said:
“Our thoughts are with the woman’s family at this very difficult time and we continue to offer them support.
“Our enquiries into this collision are ongoing and we’d ask anyone who can help to come forward.
“Anyone who may have witnessed this collision and has not yet spoken to officers is urged to get in touch.”
A spokesperson said: “Our condolences go to the family and friends affected by this tragic incident.
“We are working with the police to support their investigation.”
Those with information can contact Police Scotland via 101, quoting incident number 1270 of 14 May.