Started out with recruiter arranging a phone screen. First few rounds are slow and appear unorganized (they are probably working with multiple leads). Once you clear a few phone screens, they start giving you more attention and things start moving faster. Besides the initial phone screen by the recruiter, I had two technical phone interviews by two solution architects.
Questions were general technical questions around TCP/IP, databases, cloud computing, security & encryption, load balancing, etc.
Once i cleared those phone interviews, they invited me to Luxembourg for a series of face to face interviews. They provided a letter for the schengen visa, and booked hotel and return air tickets. They also reimbursed other incidental expenses after the interview. The face to face interview consisted of 5 x 45 min interviews. The recruiter shared the names and roles of each of the interviewers and the schedule well before the interview trip. Each round had one interviewer, except the last one, it had another interviewer joining on the phone (the lead interviewer asked permission if she could dial him in as his name was not initially communicated).
The interviewers were very respectful and courteous, contrary to the other comments here. I must say that I really enjoyed the interview process. They are looking for the Amazon Leadership Principles in you, read about it on their website, the recruiter will keep nudging you about it in her emails to you as it is very important. You must have real examples where you lived those principles. Don't make them up, as they would know when they dig deeper. If you do not live those principles (which are reasonable), Amazon probably isn't the place for you.
There was a lunch break between the the 5 rounds of interviews, and the hiring manager usually takes you out for lunch. In my case the hiring manager was sick and he did a video conferencing interview and apologized for not being able to join me for lunch.
They take notes during the interviews, and they will tell you that. One of the interviewers did not work for AWS, but for another Amazon subsidiary. This is apparently a normal part of the process - that person would ask mostly behavioral questions.
Once you finish the interviews, they tell you within a few days if you've made it or not, and then it is about making an offer to you + background checks. I have seen that they perform deeper background checks than most companies. Their offer includes a standard non-compete clause.