I applied through my university's job search engine.
My first round interview was on campus and it was two parts, one hour long. One part was a basic behavioral interview. Very straight forward - the interviewer will as for you to illustrate characteristics or qualities that are specifically important for consulting and working at ZS. Make sure you have good answers with experiences and stories to back them up (and make sure you prepare at least 2 stories deep for every common question, they may ask you "okay thats good, give me another example"). The other interview was the case interview, and ZS does it a little differently - they gave me worksheets that had their own information and objectives to solve. Once you complete one, you can move on to the next one. I really liked this because it gave you some direction and didn't just throw you in the water like traditional cases, but the tradeoff is, is that the cases are pretty difficult, and its important to not lose focus or get frustrated.
Second round is at the the closest office to your university (This does not mean you will work at that office for your internship, you have the option to choose any of the offices). The second round consisted of several rounds. First was a 45 minute behavioral - these questions are a lot more specific than the behavioral in the first round. Second round was the first case, and it was 45 mins of individual prep, and then a 45 min presentation and discussion with the interviewer. Next was another case interview, followed by a demonstrational interview, where they walk you through something similar to what you would be doing at ZS and ask you questions along the way to see how you think. Finally the last part was a lunch interview, where you ask them questions about the firm. Overall the second round was pretty exhausting, but it was a lot of fun.