The closest thing to this would be at the Liberalia, a festival in ancient Rome indicating the transition from boyhood to man. It occurred sometime in March, and a Roman boy would celebrate it when he was around 14. The boy would take off his bulla(a necklace given to him when he was a child to ward off evil), as well as his toga praetexta, then he would put on the toga virilis indicating his transition to manhood.

Source: The Private Life of the Romans by Whetstone Johnson

As others have said, this isn't really a straightforward question to answer, and the answer differs for both boys and girls.

Boys would go from being children, to youths, to adults. In donning the toga virilis a Roman didn't really become an adult. He transitioned from being a boy into being a youth. He would gain some rights but he'd still be under the patronage of his father or other older men and wouldn't have many responsibilities. In this period of youth, young Roman men were stereotyped as being badly behaved and frequent visitors of the sort of red light district area of Rome which was known for its violence and sexual promiscuity. This was a very common thing and was almost expected of young Romans. This period was viewed as a time for learning and character building and happened between the time the boy took up the toga virilis (14ish) until some time in his mid twenties when he would likely get married. Before marriage, bad behaviour or questionable activities may have been passed off as the 'rashness of youth', but once a man was married he had real responsibility and was expected to behave as such. It is certainly arguable that adulthood for men begun in their mid twenties in Ancient Rome.

For girls this period of youth didn't really exist. They were a girl until menarche at which point they became a woman because they could produce children and thus were suitable for marriage. Menarche could happen early at aged 11 or 12 but most girls would probably have married at around the age of 15.

This is a very simplified answer and I apologise for not providing many specific examples of festivals/rituals which marked these stages of a Roman's life course. I don't have any of my books handy and don't want to get my festivals mixed up. I can provide further information in the morning if anyone cares.

The takeaway from this is that it's actually difficult to say for certain at which age a boy became a man in Ancient Rome, but for girls the answer was much simpler.

Did the Romans even have a conception of adulthood like ours ? From anthropology we know many cultures had a clear adulthood ritual, something if you could accomplish would make you a man.

Did romans have something like this?

Also, the question of women, citizens/slaves, barbarians etc needs to be explorer. An interesting question indeed!

/u/memeganoob discussed the Liberalia rite of passage and how it pertains to Roman coming of age, so I'll go over the legal ramifications of adulthood in Rome as they regard property rights. As the 2nd Century AD jurist Gaius describes in his legal textbook Institutiones,

When males reach puberty, they are freed from guardianship (tutela). Sabinus, Cassius, and the rest of our teachers think that a male reaches puberty when he displays this physically, that is, when he is able to procreate. But for those persons, like eunuchs (spadones), who cannot undergo puberty, the age at which they become adults should be used. By contrast, the others of the other (Proculian) school think that puberty should be reckoned in years: that is, they judge that a male reaches puberty if he has completed 14 years ^1

Note that this institution of guardianship only applies to males who have no living father or patrilineal male ancestor (aka sui iuris). Otherwise, the person is subject to the power of his pater familias with all of his belongings being property of the household and the pater familias, in which case where he owns no personal property that would need a guardian (tutor). Nevertheless, adulthood not only provided the right for sui iuris males to manage their estates, but also for all male Romans to gain full citizenship, enabling the right to participate in the Committees and military conscription (in the early to mid republic). On the question of how to recognize the passage of adulthood, Bruce Frier and Thomas McGinn argue that Gaius's statement about the problem of judging the sexual maturity of eunuchs as well as the intrusiveness of physical inspections made age the most practical way to judge adulthood. The more pragmatic approach of using the age of 14 or 15 clearly became the dominant method, as it would be encoded in Justinian's civil law from 529 to 534.

Frier notes that while the age of majority for females was younger (12), both male and female Romans not under the power of a pater familias had similar arrangements of close supervision by a tutor, succeeded by an optional financial advisor until completely capable of managing their own affairs (Roman women did not technically join the familias of their spouses, and maintained their own estates even after marriage). For both males and females, regardless of their family status, reaching adulthood meant that they could marry and write their wills.

^1 Gaius. Ed. Frier, McGinn. A Casebook on Roman Family Law, ISBN 978-0-19-516186-1

Followup question - I read that in Ancient Rome, fathers had absolute power over the lives of their children, e.g. a father could kill his child of any age without legal repercussions.

Did this apply to children whose life was protected by virtue of their office, such as holders of Imperium, Tribunes of the Plebs, or Vestal Virgins?