Monarch – Elizabeth II
Governor-General – Sir William Slim
Prime Minister – Robert Menzies
Premier of New South Wales – Joseph Cahill (until 22 October), then Robert Heffron
Premier of South Australia – Sir Thomas Playford
Premier of Queensland – Frank Nicklin
Premier of Tasmania – Eric Reece
Premier of Western Australia – Albert Hawke (until 2 April), then David Brand
Premier of Victoria – Henry Bolte
Governor of New South Wales – Sir Eric Woodward
Governor of Queensland – Sir Henry Abel Smith
Governor of South Australia – Sir Robert George
Governor of Tasmania – Thomas Corbett, 2nd Baron Rowallan (from 21 October)
Governor of Victoria – Sir Dallas Brooks
Governor of Western Australia – Sir Charles Gairdner
26 January (Australia Day) – Darwin was granted city status
12 February – The Melbourne outdoor performance venue the Sidney Myer Music Bowl is officially opened by Prime Minister Robert Menzies.
15 February – American evangelist Billy Graham begins a tour of Australia.
February – major floods in Queensland
March – formal construction of the Sydney Opera House began
4 June – the Soviet embassy in Canberra was reopened. It had been closed since 29 April 1954 as a result of the Petrov Affair
29 July – Qantas launched its first jet service from Sydney to San Francisco via Nadi and Honolulu.
August and September – Princess Alexandra toured Australia
September – Australian National University building an "Atom Smasher"
23 September the M/S Princess of Tasmania Australia's first passenger Roll-on/roll-off diesel ferry makes maiden voyage across Bass Strait.
November – Donald Bradman batted for 15 minutes in a demonstration. He retired from cricket.
Arts and literature
William Dobell wins the Archibald Prize with a portrait of Dr Edward MacMahon
The Big Fellow by Vance Palmer wins the Miles Franklin Literary Award
On the Beach starring Gregory Peck and Ava Gardner was shot around Melbourne. The film was based on the novel by Nevil Shute.
Six O'Clock Rock screened on the Australian Broadcasting Commission's channel. It was compered by Johnny O'Keefe and was the ABC's response to Bandstand on Channel Nine.
Adelaide's first television station, NWS-9, begins broadcasting on 5 September
Cricket
New South Wales wins the Sheffield Shield
England toured Australia in the summer of 1958/59 for The Ashes; Australia won the series 4-0.
Football
Brisbane Rugby League premiership: Northern Suburbs defeated Brothers 24-18
New South Wales Rugby League premiership: St. George defeated Manly-Warringah 20-0
South Australian National Football League premiership: won by Port Adelaide
Victorian Football League premiership: Melbourne defeated Essendon 115-78
Golf
Australian Open: won by Kel Nagle
Horse Racing
Regal Wench wins the Caulfield Cup
Noholme wins the Cox Plate
Fine and Dandy wins the Golden Slipper
Macdougal wins the Melbourne Cup
Motor Racing
Jack Brabham wins the 1959 Formula One Drivers' Championship
The Australian Grand Prix was held at Longford, Tasmania and won by Stan Jones driving a Maserati
Tennis
Australian Open men's singles: Alex Olmedo defeats Neale Fraser 6-1 6-2 3–6 6-3
Australian Open women's singles: Mary Carter defeats Renee Schuurman 6-2 6-3
Davis Cup: Australia defeats the United States 3-2 in the 1959 Davis Cup final
Yachting
Solo takes line honours and Cherana wins on handicap in the Sydney to Hobart Yacht Race
29 January – Nick Xenophon, lawyer and politician
31 January – Anthony LaPaglia, actor
8 February – Andrew Hoy, equestrian rider
12 February – Sigrid Thornton, actress
24 February – Mike Whitney, cricketer
5 March – Colleen Noonan, mother
25 April - Jennifer Margaret Le Cussan, Botanist
5 June – Mark Ella, rugby union player
21 July – Paul Vautin, rugby league footballer, coach and media personality
26 July – Gary Honey, long jumper
29 July – Gene Miles, rugby league footballer of the 1980s and 1990s
12 August – Kerry Boustead, rugby league footballer
19 August – Rodney Adler, businessman
30 August – Mark "Jacko" Jackson, footballer and actor
12 September – Brad Dalton, basketball player
6 October – Robyn Maher, basketball player
11 October – Wayne Gardner, motorcycle and touring car racer
24 October – Rowland S. Howard, musician (died 2009)
4 September – Kevin Harrington, Australian actor
13 November – Anne Manning, racewalker
1 December – Wally Lewis, rugby league footballer and coach
12 December – Christine Stanton, high jumper
15 December – Greg Matthews, cricketer
22 February – Harold Hardwick (born 1888), freestyle swimmer
20 June – Sir Ian Clunies Ross (born 1899), scientist
8 August – Albert Namatjira (born 1902), Aboriginal artist
19 September – Arthur Hennessy (born 1876), Australia's first rugby league captain
14 October – Errol Flynn (born 1909), actor (died in Canada)
14 October – Jack Davey (born 1907), radio comedian and quiz show host
22 October – Joseph Cahill (born 1891), Premier of New South Wales (1952–1959)
10 November – Gertrude Bodenweiser, choreographer
11 November – Charles Chauvel (born 1897), filmmaker
24 November – Dally Messenger (born 1883), rugby union and league footballer
18 December – Edouard Borovansky (born 1902), Czech born ballet dancer and choreographer; founder of the Borovansky Australian Ballet
