Dragnet, created by and starring Jack Webb, ran from 1951 to 1959 after earlier popularity on American radio. Over succeeding decades the series was remade three times - the 1960s, the 1980s and the 2000s. It was a very realistic series for its time, with many of the stories being pulled from the files of the Los Angeles Police Department. Often, some of the people involved in the original cases were given walk-on parts in their story.
Here's the intro to the original TV series.