Background
Melsteð was the son of Jón Melsteð.
Melsteð was the son of Jón Melsteð.
He graduated from Reykjavík"s Lærði skólinn in 1882 and completed his master"s degree in history from the University of Copenhagen in 1890.
He wrote articles and books on Icelandic history. He remained in Copenhagen thereafter. He was an assistant at the Danish National Archives 1893-1903 and a fellow of the Arnamagnæan Institute for over twenty years.
In 1904 he obtained a grant to write a history of Iceland.
Along with other Icelandic scholars in Copenhagen, he founded Hið íslenska fræðafélag í Kaupmannahöfn in 1912. He was its president until his death, and the editor of its journal from 1916.
By the time of his death, the Society had a fund of 70,000 kroner. He owned a large book collection which was bought by the Brotherton of the University of Leeds, partly through the ministrations of East. V. Gordon.
Bogi donated the majority of the proceeds to a scholarship for Icelandic historians at the university.
He was Member of the Icelandic Parliament (Althing) for Árnessýsla from 1892–1893. As a member of parliament in 1893, he argued successfully that Iceland"s top official must reside there rather than in Denmark.