I interviewed at BCG London as a PhD in November-December 2014.
First round: 1 case interview (45min) 1 written numeracy test (45min)
Second round: 2 case interviews (45min each) rather quantitative
Final round: 2 partner interviews (45 min each), more focus on business understanding, communication, career motivations.
My BCG case interviews were structured as follows: 15 min of "fit" questions (focussed mainly on discussing examples of leadership and teamwork), 25 min of a case study (expect to be given exhibits at some point in the case in the first two rounds, as in an interviewer-led case), and 5 minutes to ask questions. All cases presented are based on actual engagements your BCG interviewer participated in. They thus know every aspect of the case, which makes the interview very interesting, and it also means you might want to ask your interviewer about how her or his team approached the problem at the time.
The recruitment staff was very dedicated, professional, friendly and helpful. Interviewers were friendly too, and very sharp as you would expect. Minutes before your interviews, you will given an enveloppe giving a little bibliography of your interviewer. This is the occasion to anticipate on the question you might want to ask her or him, more than trying to guess what your case will be about.
In terms of average success rates, I would say that about 7% of applicants receive an invitation to the first round, about 50% of them make it to the second round (which I found surprising), and another 40% are selected to the final round. At this stage, this means that about 80-90 candidates remain in the game, which is less than 2% of the initial application pool.
In the final round, it seems that if you are brilliant in the two interviews you are basically offered a position (hence there is no quota applied as in previous rounds), so the number of offers varies, but it seems to be around 25-35 offers in the recent years (in London).