During a career fair on campus at WPI I talked to some engineers from ExxonMobil. As a materials scientist with a PhD, I didn't expect to get anything out of it since they are mostly interested in chemical engineers but surprisingly I received a phone call two days later by an ExxonMobil recruiter to schedule an interview, on campus.
So, I accepted and three days later I was interviewed by Exxon's representative on campus. The interview was not based on technical questions at all. 100% behavioral questions, very much expected...Tell me about a time when you faced a challenging problem, how did you deal with it...what are the achievements you are mostly proud of...tell me about a time when you worked with other people in a group and you had to do something important to finish a project...stuff like that...
I think it went very well and if you are prepared to answer this questions you should not have any issues. The interviewer is very friendly and you will feel comfortable.
This was very recent, about a week ago so I am waiting for a phone call to see if they will call me for the next round. Usually, from what I have read, in the second round they will fly you to an Exxon's location for an on site interview.
That's all I have - I'll update if I make it to the second round.