At UCLA, there is the [GSO (Graduate Student Outreach) seminar][1], which features talks from graduate students (both pure and applied). The talks are (as far as I am aware) almost never research oriented. This is because speakers are encouraged to make their talks accessible to graduate students from all research groups, which makes the talks less advanced and specific in practice. Speakers often present on topics which are wholly unrelated to their own work or area. One can get a feel for the kind of talks that are given by clicking the link above and perusing some of abstracts. The seminar runs weekly, provided there is someone available to speak, which is typically the case. 

I don't think there are any rules prohibiting faculty from attending, but regardless, it is almost exclusively attended by graduate students, with postdocs and undergraduates occasionally appearing among the audience members. It was common for pizzas (or some other kind of snack) to be ordered using departmental support. Afterwards, it wasn't uncommon for some subset of the students attending to go out together. I really enjoyed these seminars when I was a grad student, and my impression is that the GSO seminar is broadly well-liked as an event. 


  [1]: https://secure.math.ucla.edu/seminars/show_quarter.php?t=1624950000&type=GSO&id=&tba=false