munus

Munus, muneris is neuter, third declension, stem muner-. Three senses that look unrelated in English are one idea in Latin: what you owe. A munus is a duty or a public office; the gift or service you are obliged to give; and, at Rome, the gladiatorial show a magistrate paid for out of his own pocket, which is why munus becomes the ordinary word for the games.

The negative tells you the sense clearly. Immunis is free of obligation, hence English immunity, and municipium is a town whose people took on Roman duties. Municipal, remuneration and common (com-munis, sharing the burden) all come from the same root. It has nothing to do with munio, "I fortify", despite the overlap in the perfect forms.