This is obviously a very broad question but here's /u/Steelcan909 answering in the specific context of murder in 1097 London https://www.reddit.com/r/AskHistorians/comments/hu4lnj/im_an_average_guy_living_in_london_in_1097_one/

I have a couple of previous answers that might be helpful:

Prior to DNA evidence, finger prints, etc. how did they solve murders and actually know if they convicted the right person?

A 12th century peasant is raped, and there are no other witnesses. How likely is she to get any kind of justice? What circumstances would make it more or less likely?

There usually were no police forces. There wasn't really a "state" to investigate and prosecute crimes. Sometimes there were courts and judges, but if someone committed a crime against you, you (or your family) were responsible for bringing them to the court.

There were sometimes institutions that were something like "police". In the medieval Muslim world there was the "shurta", which had duties that might be recognizable if we're comparing it a modern police force. The shurta would act as riot control, a sort of paramilitary force to hunt down organized bands of thieves, and maybe keep order in the courts and markets. But they also had other duties that a modern police force doesn't, like keeping the streets clean. Saladin, the sultan of Egypt and Damascus, had been the "sahib ash-shurta", the chief of police in Damascus, when he was younger.

The shurta probably influenced the Europeans who lived in the crusader states as well. The crusaders also had street-cleaners and guards who patrolled the cities at night. But in Europe there usually wasn't any sort of police force like that.

This all has to do with high medieval England, mainly.

There was something called the hue and cry, where you basically shouted alarms, and every able-bodied man in earshot was obligated to try to apprehend the criminal.

Knights and nobles could serve as a roaming justice in eyre. Their traveling court would inspect landholdings in various shires, and it was high treason for them to be killed. Sometimes locals heard that the itinerant court was coming, so they ran away. Sometimes an accused sat in prison for too long while waiting for them to show up for the trial, so they were released.

The royal forest had officers known as foresters and woodwards to stop people from killing deer, gathering wood, etc. They would perform investigations, and there's a record of them taking things like bloody arrows as evidence.