Which Renaissance banking family ruled Florence?

The Medici, who ran Florence for most of three centuries while for long stretches holding no formal office at all - power exercised through debt, patronage and marriage rather than through a title, which is a large part of why the family is studied by political scientists as much as by art historians.

The bank was the foundation and it was an innovation. The Medici ran branches across Europe organised as separate partnerships, so a failure in Bruges could not bring down Florence, and they developed bills of exchange that moved money internationally while stepping around the Church's prohibition on charging interest - a currency-conversion fee is not usury if you word it correctly.

The art everyone knows was purchased with that money and partly to launder its reputation. Cosimo funded churches and libraries in a period when banking was regarded as spiritually dangerous; his grandson Lorenzo the Magnificent supported Botticelli, Michelangelo and Leonardo. The family produced four popes and two queens of France, and the bank eventually collapsed - through bad lending, political loans that were never repaid, and branch managers who ignored head office.