Lossiemouth is a town in Moray, Scotland. Originally the port belonging to Elgin, it became an important fishing town. Although there has been …Lossiemouth is a town in Moray, Scotland. Originally the port belonging to Elgin, it became an important fishing town. Although there has been over 1,000 years of settlement in the area, the present day town was formed over the past 250 years and consists of four separate communities that eventually merged into one. From 1890 to 1975, it was a police burgh as Lossiemouth and Branderburgh.