• Monarch – Frederik IX^[1]
• Prime minister – H. C. Hansen

• 7 January – MS Hans Hedtoft departs from Copenhagen on her maiden journey to Greenland.
• 14 January – M/S Hans Hedtoft arrives at Julianehåb.
• 30 January – M/S Hans Hedtoft sinks on the way back to Denmark. 95 lives are lost.

Date unknown edit

• Palle Lykke Jensen (DEN) and Kay Werner Nielsen (DEN) win the Six Days of Copenhagen six-day track cycling race.

January–March edit

• 14 January – Lars Høgh, footballer
• 18 January – Bjarne Henriksen, actor
• 25 February – Poul Erik Tøjner, art historian and museum director

April–June edit

• 2 April – John Lauridsen, footballer
• 2 May – Lone Scherfig, film director
• 22 May – Kenneth Brylle, footballer
• 5 June – Jesper Worre, cyclist

July–September edit

• 22 August – Pia Gjellerup, politician
• 20 September – Mona Juul, politician

October–December edit

• 8 October – Erik Gundersen, speedway driver
• 13 November – Lene Hau, physicist
• 27 November – Pernille Sams, real estate agent, lawyer and politician
• 26 December – Hans Nielsen, speedway rider

January–March edit

• 12 January – Edvard Eriksen, sculptor, creator of the Little Mermaid statue (born 1876)

April–June edit

• 13 April – Dagmar Hansen, cabaret singer and stage performer, Denmark's first pin-up girl (born 1871)
• 21 April – Hakon Andersen, organist and composer (born 1875)
• 30 April – Daniel Andersen, composer, sculptor and ceramist (died 1885)

July–September edit

• 8 August – Harald Christensen, wrestler (born 1884)
• 18 August – Johan Ankerstjerne, cinematographer (born 1886)
• 28 August – Kay Bojesen, designer (born 1889)
• 2 September – Sigvart Werner, amateur photographer famous for his artistic landscape photographs (born 1872)
• 20 September – William Scharff, painter (born 1886)

October–December edit

• 9 December – John Olsen, film producer (died 1885)
• 13 December – Ingrid Vang Nyman, illustrator (born 1916)

• 1959 in Danish television