Should I join a professional society like ASME or SHPE? What are the advantages?

I can only comment on SHPE as that's the only one I'm affiliated with. I don't know how active SHPE is at your school, so I only talk about the things that should be available to all members nationally. First one is the National Conference, which has a huge career fair (around 200 companies or so go, I don't remember the exact number) and has some interesting events this year, with a Nissan competition, and a hackathon both with some decent prizes, Nissan offers you an internship if your group wins I believe (you have to apply to be apart of the competition), but it's expensive to go, and it's in Seattle this year. There's also the smaller conferences like RLDC, where there's usually company tours, a career fair, and networking events, but keep in mind this one depends on your the region your SHPE chapter is in (Some regions are better than others). There's also professional chapter that you can outreach to start your network, but again it matters on your location. There's also local events with other schools' SHPE chapters, like sports events an such, which is good for meeting other engineering majors outside of your school. That's all I can think on the top of my head, if your school's chapter is activate there will be way more benefits, but that's the bare minimum.

/r/EngineeringStudents Thread