Bogi Thorarensen Melsteð (born Klausturhólar 4 May 1860, died 12 November 1929) was an Icelandic historian.
He graduated from Reykjavík's Lærði skólinn in 1882 and completed his master's degree in history from the University of Copenhagen in 1890.
As a member of parliament in 1893, he argued successfully that Iceland's top official must reside there rather than in Denmark.
He wrote journal and newspaper articles on Icelandic language, history, education, the co-operative movement and governance.
He owned a large book collection which was bought by the Brotherton Library of the University of Leeds, partly through the ministrations of E. V.