Submitted CV and Cover Letter, was invited for a First Round interview comprising a 40 minute numerical test (negatively marked) and 2 45-minute case study interviews (with 10 minutes of fit questions). Got called back for the Final Round, which was made up of 3 45-minute case study interviews (again with 10 minutes of fit questions). Heard back within 3 days.
They told me that fit was as important, if not more important than the case study itself. From my feedback after the first round, my interviewer mentioned that they really like it when you go in-depth for the fit questions: they want to know how you think, why you did what you did etc.
Obviously, as many others have mentioned, doing maths accurately and methodically within the case is important, as is having a good structure from the beginning. I had terrible structure when I first started interviewing with other firms, and I spent a lot of time practicing case starts (timed), so that I was able to tailor the structure to the exact question, which is precisely what BCG looks out for. Also try not to come up with a complicated structure to impress them, just make sure you don't force-fit. All my structures were very simple capability vs. profitability/ cost vs. profit types, but they covered all the bases.
The people at BCG were extremely friendly, and they seem to really look out for evidence of teamwork, people skills, and creativity/ desire to learn. It definitely pays to read BCG perspectives beforehand as well, to gain an insight into how they crack cases and their style of consulting (which, as they mention, is very tailored to the client's situation, and very data-driven).