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Most medieval kings had a court system spread throughout their lands. They maintained "High" Courts for issues of national importance.
For courts to decide "local" matters, they maintained local magistrates. In the case of England, these included sheriffs in cities, and justices of the peace, who were country squires. Other European countries had similar structures. In many cities, there were guild courts maintained by trade guilds, who decided cases of economic crimes.
Upvote:7
Town charters were a development of the High Middle Ages, and defined the privileges enjoyed by towns in Europe. Often, these charters explicitly established law courts to allow legal disputes to be settled quickly. Charters would also usually establish trade guilds in towns which maintained their own system of courts.
Before that, the responsibility for settling legal disputes lay with the sheriff, or shire reeve, who was the king's representative in a city, town or shire. The reeve was responsible for collecting taxes and enforcing the law locally.