I'm sure a specialist in social history can be more helpful here, but urban Rome was a relatively lawless place. There were constant fires, night was regarded a very dangerous time to be on the street, there was no police force and lots of criminals. Protection at night came from weapons and company, just like it would today in a dangerous city.

As there was no police force no common crime would be formally investigated. However, as would be expected in any society, people had family and friends and other contacts who could help them if they or someone they knew was a victim of crime (especially the client relationships that connected almost everybody in Rome, no matter how distantly, with an actually powerful person). Some Collegium were also like proto-mafia criminal gangs that could probably help find things out and get revenge for a price.

Serious crime in the ruling classes was a totally different matter. Aside from the very public political violence that was a major factor in the collapse of the Republic, which by its public nature didn't really need much investigation (and on numerous occasions resulted in the execution of citizens without trial, an anathematic concept to the Republican Romans), moral and physical crimes in the aristocracy under various circumstances could be framed as serious public issues. Examples include Clodius sneaking into a girls only religious ceremony in a private household in the time of Caesar or the deflowering of a Vestal Virgin. When Augustus recreated the state, it was on strong moral and physical authority and any challenge to that, whether it be from excessive criticism from politicians or the sexual misconduct of his daughter, was investigated and punished severely.

After Augustus political crime took a different dimension. The Emperor was the state, and any threat to his authority was a threat to Rome. Such threats abounded internally, especially early on in the Empire or when an Emperor's authority waned for some reason or another. From Augustus the use of informants and specialists that could almost be called secret police became normal practice in the Empire, especially Rome itself. Maiestas, or treason, became perhaps the most serious crime of all and was sniffed out thoroughly by imperial servants and punished severely

My main sources off the top of my head are Daily Life in Ancient Rome by Jerome Carcopino and SPQR by Mary Beard (not really an academic source despite its author's pedigree). I think Beard references some primary source material in the form of Roman literature or poetry that makes light or complains of the dangers of Rome at night, but I can't say who these sources are as I've never read them. Any decent book on Augustus or the early Empire, like Eck's "Age of Augustus" should give an outline of political crime in the Principate.

As there was no police force no common crime would be formally investigated.

Otherwise good answer from u/PapurusCursos and I generally agree that the Roman Empire was largely a self-help society, but I'll have to disagree with this point. The Romans DID have institutions of law enforcement in place and were indeed 'deeply engaged in maintaining public order' as Christopher Fuhrmann showed in his newish book (review). If you were wronged, there were different authorities you could turn to - in the provinces, this was different military officials solely or almost solely concerned with policing, such as the beneficiarii, stationarii, regionarii and frumentarii, or then you could petition to local magistrates. We have hundreds of papyri left from Roman Egypt where the inhabitants (note - not even just citizens, just all subjects of Rome) petition to the Roman authorities to redeem injustices they have suffered. The reported crimes range from illegal pasturing to stolen olives, from verbal insults to physical abuse, rape, or even murder. The authorities are often asked either to catch and bring to justice a named wrong-doer, or sometimes to investigate when the identity of the criminal was unknown, as was often the case with thefts.

Unfortunately, these petitions, while extremely interesting documents, have quite clear limitations when we try to assess just how effective the law enforcement structure was. Since papyri pretty much only survive in the dry climate of Egypt and nothing like them has been found almost anywhere else in the Roman world, we can't say just how well the policing institution of Egypt (in many respects a highly unique and peculiar province) was representative of the whole Empire. Just because a subject applied for help from the authorities does not mean that he or she received it; and although some court proceedings and correspondence of on-going or finished police/law cases survive from Egypt, too, the data is just not big enough and too random (in terms of geographic locations, time spans etc.) to make any conclusions about how common it was to submit a petition and then be successful in getting justice in court. Also, we don't know what went on in Roman Empire policing outside of the Egypt petitions, like, what these policing officers could (or would) do in the field without a formal petition or orders from higher officers. The picture that emerges from the Egypt petition is that they could intervene spontaneously if public disorder arose, and even help individuals in informal dispute solution, hold potential plaintiffs that tried to escape etc.

So, back to u/Bears85's question. The Empire did not have a 'public prosecution' system that chased criminals that had offended private persons; unless someone petitioned to investigate the murder you've commited, you'd get away. But, let's imagine that this happens and look at this question from the victim's perspective. What would you do, if you, say, wake up to find that your son has been murdered, robbed and killed during the night?

You would go to the authorities. We don't know yet why the Romans petitioned to any particular authority in any particular instance - the local magistrate, governor of the province, or army officials were all just as likely to receive petitions about illegal pasturing or violent assault. It's quite likely that your personal relations and the accessibility to any official were among the biggest factors. Anyway, you decide that the best person to petition to is the local army centurion, because a reply from the governor might take months (showing that the law enforcement system did have substantial weaknesses!). You go to the village scribe to get a formulaic petition written - the fees of the scribes were modest, so unless you were among the poorest of the poor, you could afford to petition even if you were illiterate. In the petition, you describe what has happened - it's also always good to add some heart-wrenching emotional pleading, e.g. "my only son and safety for my old days has been cruelly robbed from me" - and request that the centurion starts investigating the murder. After a few days, the centurion receives the petition and decides that it is convincing, and sends some of his policing soldiers to investigate.

Now, what would this investigation entail? Unfortunately, we know hardly anything about the process. But, from court proceedings and orations, we know that the evidence brought up in court consists basically completely of witness testimonials, accounts, or letters - no hairs, footprints or anything that would point to some sophisticated forensics methods. So, the military officials would probably ask you who could be guilty of your son's murder: did your son have any enemies? Have any slaves disappeared or acted suspiciously? Is there anyone, who witnessed the crime? If the police officials cannot find any suspects through questioning, that would probably be the end of it.

But, say, it happens that the soldiers do find a suspect - your son's friend was suspicious of their common friend who left with him after a night in the brothel, someone else has seen him wearing bloody clothes on that same night, and this same friend has suddenly disappeared from town. The Romans had surprisingly efficient systems in place for catching fugitive slaves - it's unclear, though, if the same systems were used to catch private offenders on the run and just how difficult it was to live as an 'outlaw' in Rome. Most likely, in a time before photographs and as long as you were not a public figure: not that difficult at all.

But, so we can answer your last question, let's imagine the soldiers were able to catch this murderer and prove his guilt in court before jury (in Roman court, establishing guilt 'beyond reasonable doubt' was not always necessary - it was sometimes enough for the prosecution to prove that he is the kind of person who would do this!). What would the murderer's punishment be? Well, the Roman law system was differential - where on the social pyramid the murderer and the murdered stood made all the difference (citizenship status, wealth, birth etc.), and although the Roman laws could sometimes be astonishingly specific, the verdict was also largely dependent on the interpretation of the judging magistrate. If you murdered someone of considerably lower status, your punishment would probably be a fine of some sort, tops; if it was a slave, no punishment whatsoever (although the owner of the slave had the right to claim compensation). If you killed someone around your equal, the punishment would have been either a hefty fine or exile. If, however, you killed someone clearly above your status, you could get capital punishment - which, for a citizen, usually took the form of a beheading, which was the more honourable way to die.