Eske Brun (25 May 1904 – 11 October 1987) was a high civil servant in Greenland and in relation to Greenland from 1932 till 1964.
Brun was born in Aalborg in the northern part of Jutland, Denmark. His father died when he was 15 and the family moved to Ordrup north of Copenhagen. He received a law-degree from University of Copenhagen in 1929.
In 1932, at the age of 28, Brun was given a substitute job as governor of North Greenland. In 1939 he got a permanent position as governor. When World War II started and the connection to Copenhagen (the capital of the Kingdom of Denmark, which Greenland was a part of) was severed, on account of the German occupation. Brun and his colleague Aksel Svane, via the law concerning the government of Greenland of 1925, took control of the island. They established supply-lines from United States and Canada with the help of the Danish ambassador in Washington, Henrik Kauffmann. From 1941 until the end of the war Aksel Svane was situated in US to organize the supplies and Eske Brun became governor of South Greenland as well. The administration was centralized in Godthåb (Nuuk).
After the war Eske Brun was made vice-president of the Greenland Administration (Grønlands Styrelse) (in 1947). In January 1949 he succeeded Knud Oldendow as president. Eske Brun worked as a senior official until his voluntary retirement in 1964 after disputes concerning equality between Danes and Greenlanders.