https://mlh.io/seasons/s2016/events Is very popular for promotion.

Are hackathons just a university thing, then...?

By and large. Major League Hacking (the link above) is sort of like the national group "in charge" of hackathons and one of the requirements to be recognized by them is to be a purely under-grad event (though sometimes highschool students are admitted). That said, there are plenty hackathons that aren't sanctioned by MLH and therefore are more open.

It's also possible that my view of hackathons is skewed by the fact that I'm a university student at a school with a very active hackathon culture.

No. Not at all. The average age for our hackathons was closer to 35 than 20.

Where do you advertise them? How do people find them?

We used journo-tech communities like Hacks/Hackers, college boards, company mailing lists (ie. the post design group), or directly in touch with influencers (ie. NYTimes technical evangelist).

I ask because I feel like I keep up pretty well with tech news, reading multiple news aggregators every day - and yet I am clearly not connected to whatever information channels people are using to promote hackathons, since I never hear about them. I have no idea whether I'd want to participate in them even if I did hear about them, but it makes me aware that there must be some avenues for tech news communication that I am totally missing out on.

You can tell from the examples that these are mainly media related. This industry is pretty tight nit with preexisting networks of meetups/listservs/influencers.

I have mostly been to them through word of mouth, heard through friends, colleagues, related meetup groups, Reddit, etc.