Started the process through a referral. The process consists of three rounds, the first just being a test similar to the GMAT but not multiple choice, only math, 20 questions in 20 minutes.
Then two 45 minute interviews, going over experience and some personal details, but mostly focusing on the case studies. These are cases that BCG actually worked on and summarized in a way that you can have a general sense and provide some insight. It's important to show your reasoning as you go along, as well as ask questions to be sure that you're getting all the information.
I didn't make it to the second and final round, but from what I understood it's the same case study format, just with more senior members of the team.
Everyone I dealt with, really everyone, was extremely polite, professional, helpful and generally simply nice to talk to and deal with. The only downside I saw was that there was no feedback process, but the fact is that the competition is very tough, so even if you do well and don't really have anything in particular you need to improve, someone could very well have done better or have a more apt background or whatnot.