He is a policeman (岡っ引き, okappiki) who catches criminals by throwing coins, the zeni of the title, thus Zenigata Heiji.
Heiji's beat is Myōjin-shita, "beneath the (Kanda) Myōjin shrine".
He uses deduction, a jutte, and old-fashioned coins with a hole, called kan'eitsūhō (寛永通宝), as weapons to catch criminals.
Films about Zenigata Heiji include: The lyrics of the ending theme of the TV show changed every week depending on the plot line.
In the TV series, the closing credits show a coin, based on the kan'eitsūhō coin thrown by Heiji, with Zenigata Heiji written on it.