They were approved in 1960 with construction carried out by Laing. Construction ended the following year. The building contained 80 flats.
The decision was taken to demolish the troubled high rises in Oxgangs at the turn of the century and the building was derelict by January 2006. In August 2006, preparation for demolition commenced. Allermuir Court was demolished by Safedem of Dundee at 12 noon on Sunday the 26th of November 2006. The oldest residents of teh Comiston high rises, Winifred Muirhead and Janis Dickenson pressed the detonator. They had lived in the flats for a combined total of 70 years.
On the day of demolition, there was a small exhibition set up documenting the flats 46 year history set up by Heather Levy of Firrhill Community Council. The site was completely cleared by February 2007.