Any authority can of course claim ETJ over any external territory they wish. But for the claim to be effective in the external territory (except by the exercise of force) it must be agreed either with the legal authority in the external territory, or with a legal authority which covers both territories. When unqualified, ETJ usually refers to such an agreed jurisdiction, or it will be called something like "claimed ETJ".
Extraterritorial jurisdiction can apply internationally. For example, the United States has Status of Forces Agreements with many nations which give the United States jurisdiction over members of its military.
Many countries have implemented laws which allow their nationals to be prosecuted by their courts for crimes such as war crimes and genocide even when the crime is committed extraterritorially. For example the United Kingdom has incorporated the International Criminal Court Act into domestic law. It is not retroactive so it applies only to events after May 2001 and war crime charges can be filed only against British nationals and residents. According to Peter Carter QC, chairman of the Bar's human rights committee "It means that British mercenaries who support regimes that commit war crimes can expect prosecution".
Prevent Genocide International, "a global education and action network established in 1998 claims that crimes such as genocide need to have extraterritorial jurisdiction so that people that commit such crimes can not find protection in a country that does not have such a law:
In the U.S., many states have laws or even constitutions which permit cities to make certain decisions about the land beyond the town's incorporated limits. Texas, Alaska, North Carolina, and Arkansas are all examples of states which allow cities to claim ETJ over zoning laws, which is the most common use of ETJ. In California, ETJ is referred to as a city's "Sphere of Influence". Some examples include Los Angeles which has a sphere of influence over the unincorporated islands of census designated places in and around its boundaries, and Palmdale which has a large sphere of influence on its unincorporated suburbs as well as the cities within its U.S. government designated Foreign Trade Zone. Some states require that ETJ be established over an area for an extended time period (usually one year) before it can be subject to annexation. Most set the ETJ distance based upon a city's population.
In Australia, extraterritorial jurisdiction of the various state and territory governments and government bodies is accommodated by the Australia Act 1986, although this is only one of many purposes of the Act.