Police received multiple reports of a firearms discharge in Chester Street at around 8pm on Wednesday April 17.
Officers and other emergency services responded to the area and a man was found lying on a common staircase, having sustained a serious injury.
He later passed away at the scene.
Police say formal identification has still to take place, however, the victim is believed to be 48-year-old Bradley Welsh, who lived nearby.
This death is being treated as murder.
Early investigations indicate that this has been an isolated attack and inquiries are ongoing to trace those responsible.
Anyone with any information that could assist the investigation please come forward.
Detective Superintendent Allan Burton from the Major Investigation Team said: "At this time our deepest sympathies are with this man's family and a significant inquiry is now underway to trace everyone who was involved in the murder.
"I would ask that anyone who was within Chester Street, or the West End of Edinburgh on Wednesday evening, and who saw anyone, or anything suspicious, to contact the police immediately.
"Part of this investigation will focus on obtaining CCTV from nearby homes and businesses and we would also urge any motorists who were in the area and may have relevant dash-cam footage to share this with us.
"Murders remain extremely rare in the Capital, and such incidents where a firearm are used are even more uncommon. However, we wish to reassure the public that considerable resources are being dedicated to this inquiry and we are treating this matter with the utmost seriousness."
Chief Inspector David Robertson, Local Area Commander for the City Centre added: "We recognise and understand the profound impact this incident will have had, both on those connected to the victim, and to the local community of the West End.
"There will naturally be a high officer presence in the area over the forthcoming days both to offer reassurance and gather any relevant information that may be of use to the inquiry.
"I want to stress that Edinburgh remains a safe place to live, work and visit and we are doing everything we can to bring those responsible for this needless death to justice.
"In the coming days, weeks and months we will be providing the family with all the support and assistance they may require."
Those with information can contact Police Scotland via 101 and quote incident 3782 of the 17th April. Alternatively, an anonymous report can be made to the charity Crimestoppers on 0800 555 111.