Most countries define the age of adulthood at 18 years.
Why is this? Was there some historical precedent set in ancient Rome/Greece or something like that? Why 18 years and not some "nicer" number like 20 or 15?
Age of consent laws could, in part, explain why we've settled on 18 or ages near to it. Age of consent statutes can be dated as far back as 1275 in England and were adopted in a number of other countries throughout Europe. Some of the first interpretations settled on the "age of marriage", which at the time was 12. Part of the problem with having a set number that determines the age of consent is explained by Robertson:
In trials, juries were often unwilling to simply enforce the law. Rather than focusing strictly on age, they made judgments about whether the appearance and behavior of a girl fit their notions of a child and a victim. It was not only that relying solely on age seemed arbitrary to them; at least until the end of the 19th century, age had limited salience in other aspects of daily life. Laws and regulations based on age were uncommon until the 19th century, and consequently so was possession of proof of age or even knowledge of a precise date of birth.
Once it became possible to reliable check the age of a person, the laws demarcating age became easier to enforce. Robertson continues:
At the end of 19th century, moral reformers drew the age of consent into campaigns against prostitution. Revelations of child prostitution were central to those campaigns, a situation that resulted, reformers argued, from men taking advantage of the innocence of girls just over the age of consent. The outcry it provoked pushed British legislators to raise the age of consent to 16 years, and stirred reformers in the U.S, such as the Women's Christian Temperance Union, the British Empire, and Europe to push for similar legislation. By 1920, Anglo-American legislators had responded by increasing the age of consent to 16 years, and even as high as 18 years.
What followed in the 20th century was a debate on what constitutes physiological, and psychological maturity. No doubt that this and other factors such as: compulsory schooling, mandatory military service, liquor laws, and the rise of higher education as a intermediary between adolescence and adulthood, played a role in us collectively as a society deciding that someone is an adult at 18.
Stephen Robertson, "Age of Consent Laws," in Children and Youth in History, Item #230, https://chnm.gmu.edu/cyh/case-studies/230 (accessed May 26, 2016).
This paper by T.E. James (published in 1960 so it is a bit dated) suggests that common law settled on 21 as the age of adulthood. Common law was influenced heavily by Roman and Justinian laws, though these generally settled on 15 as the age of majority. James write:
In the absence of any clear authority, it may be assumed that at one time between the ninth and tenth century, fifteen was also the age of majority in England; but, by the time of the Magna Carta, this age had been raised to twenty-one, at least so far as men holding in knight service were concerned.
I think some have mentioned that age might have been linked to the increased weight of armor. I think there seems to be some truth in that. The fact that James writes in 1960 that the age of majority is 21 and that it has been that way for centuries (at least in England and some other European countries) lends itself to the argument that 18 might be a relatively recent age change.
For a long time, 21 was the "age of adulthood". From Encyclopedia of Adult Development:
As early as the thirteenth century, age 21 was an important marker in Britain, at least for men. This was the age, for example, when they could begin to serve as knights. It was believed that the physical requirements of combat - to wear a heavy suit of armor and simultaneously lift a sword or lance - could only be met by men who had attained age 21.
It wasn't until the 26th amendment in 1971 that the voting age in the United States was lowered from 21 to 18. A big reason for this was the outcry that men were "old enough to fight, but too young to vote".
I would guess that the reason men were drafted into the military at age 18 was because of the need for as many soldiers as possible and since armor was no longer in use, a person did not have to be as physically developed to serve as a soldier. Additionally, most men finish high school at age 18 so they have all of their primary and secondary education completed.
On a minor detail, as far as I know ancient Roman law had several stages and ages of majority.
Roman boys had a ceremony to remove the toga praetexta with a broad purple border, and their protective bulla praetexa amulets, and assume the pure white toga virilis, the toga of manhood, at 14, considered to be the age of male puberty in ancient Rome. Roman boys would have their first military service sometime in their late teens. And I believe the end of adolescence and the age of complete adulthood was 25. And the minimum age to run for elected public office was 30.
So the closest fit between ancient Roman and modern legal ages is that the minimum age for military service is similar.
There were several reasons for 18 years old being the threshold of adulthood.
The first reason was military, as embodied in the 26th Amendment. During World War II, Congress lowered the draft age from 21 to 18, more or less in line with the other major combatants. This actually became a sore point during the Vietnam War, when the slogan became, "old enough to fight, old enough to vote." And "voting" was basically the key right of adulthood.
A second reason was that until the post war period, most children completed their (secondary) education at the age of 18, and entered the work force at that time. Here, again, they had "adult" responsibilities at age 18.
A third reason is that 18 is the maximum age of "consent" for marriage, and "marital relations" (including those undertaken outside of marriage). Although some states and countries have lower ages of consent.
It's true that for most people, particularly men, intellectual development is not complete, and perhaps one is not fully adult until age 25. (This is why a Congressman must be that age.) But that's not the point, which is that an 18 year old is "more adult than not," which is why the line is set there.