The prime minister of Denmark (Danish: Danmarks statsminister, Faroese: Forsætisráðharri, Greenlandic: Ministeriuneq) is the head of government in the Kingdom of Denmark comprising the three constituent countries: Denmark, Greenland and the Faroe Islands. The kingdom did not initially have a head of government separate from its head of state, namely the monarch, in whom the executive authority was vested. The Constitution of 1849 established a constitutional monarchy by limiting the powers of the monarch and creating the office of premierminister. The inaugural holder of the office was Adam Wilhelm Moltke.

The prime minister presides over a cabinet that is formally appointed by the monarch. In practice, the appointment of the prime minister is determined by their support in the Folketing (the National Parliament). Since the beginning of the 20th century, no single party has held a majority in the Folketing so the prime minister must head a coalition of political parties, as well as their own party. Additionally, only four coalition governments since World War II have enjoyed a majority in the Folketing, so the coalitions (and the prime minister) must also gain loose support from other minor parties.

The current prime minister of Denmark is Mette Frederiksen, since 27 June 2019. Since 2022 her cabinet consists of the Social Democrats, Venstre, and the Moderates.

From approximately 1699 to 1730, the highest-ranking non-monarchial government official was titled "Grand Chancellor" (storkansler) and from 1730 until 1848, this office was titled "Minister of State" (Statsminister). These titles foreshadowed the modern office of prime minister, however, unlike the current office, the grand chancellor and state minister were not formal heads of government. The king held executive authority as absolute ruler from 1661 until the enactment of a liberal Constitution in the early nineteenth century.

The office of prime minister was introduced as a part of the constitutional monarchy outlined in 1848 and signed as the Danish Constitution on 5 June 1849. The new Constitution established a parliamentary system by creating a new bicameral parliament (Rigsdagen) and a Council Presidium, headed by a council president. The Council Presidium is regarded as the predecessor of the modern Prime Minister's Office. The first council president was Adam Wilhelm Moltke, who came to power on 22 March 1848. Molte and his next two successors also held the title of premierminister, which translates as "prime minister".

From 1855 onwards the prime minister was known simply as the "council president" (Konseilspræsident). Carl Christian Hall became the first prime minister/council president to lead a political party (the National Liberal Party).

The modern Prime Minister's Office was founded on 1 January 1914, when the Council Presidium was established as a department under the prime minister, when it had previously existed as an informal council gathered by the prime minister. The title of the prime minister changed again in 1918 under the premiership of Carl Theodor Zahle, becoming titled Statsminister or "Minister of State", in line with its Scandinavian neighbours, Norway and Sweden), which it remains to this day.

By the mid-nineteenth century a strong party-system had developed, with most prime ministers being the leader of either Venstre (left) or Højre (right). However, by 1924 the Social Democrats had become the largest party and Højre had disappeared.