Acting prime minister

An Acting Prime Minister is a cabinet member (often in Westminster system countries) who is serving in the role of Prime Minister, whilst the individual who normally holds the position is unable to do so. The role is often performed by the Deputy Prime Minister (where that position exists), or by another senior minister. 
The office is commonly used when the Prime Minister is absent from the territory of that nation or when the Prime Minister is unable to perform duties for health reasons.
An Acting Prime Minister should be distinguished from a Caretaker Prime Minister, which refers to an Outgoing Prime Minister following an electoral defeat, and who by convention does not implement new policies or an Interim Prime Minister who is appointed to perform a similar role to a Caretaker Prime Minister, but who is typically not a Prime Minister at the time of being appointed.
Statutory authority.
Within a Commonwealth nation, the office of Acting Prime Minister like that of Prime Minister is found only in convention, and is not legislated. The Department of Prime Minister and Cabinet, or an equivalent ministry, drafts, reviews and publishes any formal conventions (though these are expressly non binding in a court of law).
By country.
Australia.
According to "House of Representatives Practice", an official publication of the Parliament of Australia, the Prime Minister of Australia "may make temporary ministerial arrangements without reference to the Governor-General. A Minister may act for another Minister on account of absence from Australia or from the Ministry or due to ill health. The "Acts Interpretation Act" confers upon an Acting Minister the same power and authority with respect to the absent Minister's statutory responsibilities."
The position of "acting prime minister" is a special case of an acting minister, and generally occurs when the prime minister is travelling overseas, is on vacation, or is in ill health. In the early 20th century, when travel by ship was still the norm, it was not uncommon for there to be an acting prime minister for months on end. 
The deputy prime minister is usually designated as the acting prime minister, although another senior member of the government may fill the role if both the prime minister and deputy prime minister are unavailable. An acting prime minister is required quite frequently – for instance, between 3 December 2007 and 23 February 2009 (during Kevin Rudd's first term), Julia Gillard acted as prime minister on 16 separate occasions.
The term "acting prime minister" is sometimes also applied to someone who is temporarily appointed Prime Minister following a death in office (more commonly called a "caretaker prime minister" or an "interim prime minister"). Examples include Frank Forde (seven days following John Curtin's death), Earle Page (19 days following Joseph Lyons' death), and John McEwen (22 days following Harold Holt's disappearance). 
Unlike those who merely acted in the absence of a prime minister, Forde, Page and McEwen were officially commissioned by the Governor General and took the oath of office; they are considered prime ministers in their own right.
Canada.
In Canada, the Deputy Prime Minister—a position that is not always in use—becomes acting head of government in the event that the Prime Minister is absent from Canada or incapacitated. Prior to the creation of the Deputy Prime Minister position, acting prime ministers were routinely appointed when prime ministers were temporarily away from their duties; however, prime ministers typically appointed a different cabinet member each time, rather than designating one specific person as a permanent official "deputy". 
In February 1958, Prime Minister John Diefenbaker appointed Ellen Fairclough as acting Prime Minister for two days while he was absent from his duties; Fairclough was historically noteworthy as the first woman ever given the duty.
China.
After the death of Chinese First Premier, Zhou Enlai, Hua Guofeng served as acting Premier of the State Council for two months before he took over as Premier.
Israel.
The designated Acting Prime Minister takes the role of Prime Minister as Acting Prime Minister, for up to one hundred consecutive days, if the incumbent is temporarily incapacitated. Whilst in other countries the term "Acting Prime Minister" only refers to an individual actually performing the role, in Israel the term is also in use when a designated Minister is allocated, even if they never actually perform the role.
New Zealand.
In New Zealand, the Acting Prime Minister is the designated head of government when the incumbent is not available to fulfill their duties. This includes when the Prime Minister is overseas, unwell, or otherwise temporarily unavailable. Typically this position is filled by the Deputy Prime Minister. Hugh Watt, who was then the Deputy Prime Minister, served as Acting Prime Minister for six days, after Norman Kirk died in office in 1974. 
Deputy Prime Minister Winston Peters was serving as Acting Prime Minister as of 21 June 2018 until 2 August 2018, while Jacinda Ardern took maternity leave.
Prior to the establishment of the Deputy Prime Minister role, a senior Minister would be appointed to act as Prime Minister when, for example, the incumbent was travelling overseas. In the current era, when both the Prime Minister and Deputy Prime Minister are unavailable, the next highest ranking minister who is available will be asked to act as Prime Minister, until a colleague of higher rank is available again.
Sri Lanka.
In Sri Lanka, an Acting Prime Minister is a senior minister appointed to take the role of Prime Minister in the absence of the incumbent. Wijeyananda Dahanayake was appointed Acting Prime Minister on September 26, 1959 following the assassination of S. W. R. D. Bandaranaike. He was subsequently confirmed as Prime Minister by parliament. 
United Kingdom.
In the United Kingdom, no person has ever been formally appointed Acting Prime Minister. The Deputy Prime Minister or First Secretary of State is often considered the Prime Minister's deputy. However, the office of Deputy Prime Minister confers on its holder no automatic right to succession and other ministers who have been deemed "de facto" deputies have held other offices.
Rodney Brazier lists 14 unambiguous deputies to the Prime Minister since 1940, most of whom were never formally appointed Deputy Prime Minister. These include First Secretary of State Dominic Raab under the current Prime Minister, Boris Johnson, and Willie Whitelaw under Margaret Thatcher, until 1988.
When the Prime Minister is travelling, it is standard practice for a senior duty minister to be appointed, who can attend to urgent business and meetings if required, though the Prime Minister remains in charge and updated throughout.
On 6 April 2020, when Johnson was admitted into ICU, he asked Raab "to deputise for him where necessary", though Raab was not appointed Acting Prime Minister.