Is it typical for a company to run the busses? Somewhat interesting as that's one thing in the US that is rarely privatized. The driver, etc. will be a direct employees of the municipality.

It will often be a company owned by the state.

In Ireland, most buses are run by CIE, a state company. In Dublin, for the last couple of decades, some routes have been run by GoAhead, a private company (Dublin Bus (part of CIE) and GoAhead tender for the routes). It's been a fascinating experiment in privatisation; the buses all use TFI livery, so the only way to tell whether your route is run by Dublin Bus or GoAhead, besides looking it up, is that if it's Dublin Bus, it'll just be late; if it's GoAhead it won't show up at all. For all that people complain about public sector inefficiency, private companies turn out to be very, very bad at competing with the public sector.

It's usually a company owned by the city or the state, e.g. https://en.wikipedia.org/wiki/Berliner_Verkehrsbetriebe

in EU it's normal

- a company run the busses

- its partially of fully owned by the city/state, i.e. NOT private investors, not private liquidity, no one expects any payouts

- its commonly a special form of "company", e.g. "Anstalt des öffentlichen Rechts" in this case which you can roughly read as "installment/company based on public-law"

Practical that is "mostly" the same as state run when looking at it from a distance. But with some differences. Like it's still a company, has it's own company structure, it's own bookkeeping, it's own employees etc. This leads to a much more clean separation when it comes to budged managing, and where possible removes/reduces direct political influence into daily operations (just direct one, indirectly through ownership etc. there is very much still influence, but this makes it somewhat less suspect to damage from changing political parties non stop changing directions).

Through things can get far more complicated then just one company, in one legal form.

Depends. NYC busses used to be a mix of municipal and privatized companies that were part of the Transit Alliance. In addition to city owned and operated buses you had four private companies: Green Bus Lines, Jamaica Buses, Triborough Transit and Command. Around the mid 00s they were all absorbed into the MTA. Living in south Queens the Green Bus Lines were a big part of my youth as I took the Q41 to school every day and the Q11 on weekends to the mall.

Normally municipalities own companies that do the public transport service in Germany

They're usually private companies owned by the cities.