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My experience is atypical in a number of ways. So I'll begin my answer by describing the general ways in which mathematics graduate study at UChicago differs from many other universities. You can cross-check what I say against Uchicago Mathematics Graduate Program.No qualifying examinations: In many graduate programs, students are required to pass qualifying examinations in algebra, analysis, and topology, and many students who have already taken related courses in their undergraduate days, or self-studied the topics, take the qualifying examinations at the beginning of their graduate program, and can then begin their research work right away. The University of Chicago does not have such examinations.Long list of first-year courses that everybody is expected to take by default: UChicago follows the quarter system (like Stanford and Northwestern). There is a clearly defined sequence of courses first-year students are expected to take. This includes one course each in algebra, analysis, and topology every quarter (more information here: UChicago Mathematics Department Graduate Courses). Students can opt out of individual courses if they convince the instructor of that course that they know the material -- however, this option is generally not encouraged or recommended. Each course has (roughly) one homework per week, though some instructors have bucked the homework-giving trend. As a result, the first year is devoted almost entirely to coursework, and students have very little time for other academic or non-academic pursuits. The theory is that all graduate students should have a strong foundation in all subjects, and also that they get to know each other well to form a support system for each other in the coming years. See alsoSecond year devoted to being "College Fellows": teaching assistants to faculty members for faculty-taught undergraduate courses, mostly for math majors: The goal of the second year is to give students a thorough grounding in the mechanics of teaching and grading by having them as teaching assistants for a course taught by a faculty member, before they are ready to teach their own courses. This is in contrast with other universities where instructors directly enter their long-term teaching role directly without a training period.Topic examination: Students are expected to prepare a write-up for a topic and defend it, see Topic Guidelines -- this may not be too unusual though.Full-service teaching from the third year onward: Graduate students are appointed to the position of Lecturer for the College. That title allows them to teach full-fledged courses to undergraduates. This includes not just teaching the class but also setting the tests and allocating final grades. Even though many of them are teaching multi-sectioned courses (i.e., there are other sections with the same course number and same syllabus) instructors have some flexibility in the exact choice of subtopics and pace of coverage, and can deviate to some extent from the guidelines given by the department. There are no course monitors or coordinators for multi-sectioned courses. Thus, the experience of students taught in classes by graduate students can vary considerably. At least some undergraduate students have reported frustration with the huge degree of variability, but on the other hand, the greater variability means more potential upside.Teaching in (almost) every quarter: With the exception of people who get funding from outside sources (such as NSF fellowships) all graduate students are expected to teach every quarter (autumn, winter, and spring; the summer is off and supported through a general Research Assistantship) until they graduate, except one (i.e., they're usually given one quarter off, typically in their fourth or fifth year). This is in contrast with some places, where people are required to teach or TA one semester out of two, or two quarters out of three.My personal experience was atypical, because:There were very few people in the university working in my area of Ph.D. research, group theory (in particular, there were only a couple other graduate students working even in remotely related research areas). Thus, I wasn't a regular participant in any of the regularly held seminars (the Group Theory Seminar was held very occasionally), and my academic interactions with other graduate students (after the first year) were limited to situations where they had a group theory question or (rarer) I had a question about their area of research.Some time around my fifth year, I determined that I am not planning to continue in academia, so I didn't do many of the things that one would need to do in order to continue in academia. Thus, I don't have much knowledge of how good UChicago looks as a grad school to potential employer, etc.Below are some beliefs I've gathered through some combination of my own experience and my (limited) interaction with other graduate students:The first-year courses tend to require a lot of effort, and they've engendered a mixed response. Some students have claimed that it has been a character-building experience, and has helped them with thinking and writing mathematics as well as reading mathematical papers, even papers unrelated to the subject matter they studied. Some students have argued that the demands of coursework leave them with very little time to shop around within mathematics and make a smart decision of their area of specialization. Some students have also claimed that the repeated act of writing down solutions at the last minute that somebody else has discovered (and that you only very barely understand, and couldn't reproduce on your own a few days later) breeds cynicism about the mathematical enterprise. I believe that the faculty who lay out the graduate program have made some changes in response to the feedback, but I'm not sure where things stand right now.There were complaints in the past from students that they didn't have a clear idea of what exactly the faculty members do, and which ones are open to taking students. The department recently started organizing a Meet The Faculty Colloquium (active in the Spring Quarter) to fix this issue.The first-year courses really do help create a significant bonding between people in the same year, compared to how well people from different years know each other. This can be very helpful for finding a support network in future years for teaching and research and non-academic work.The use of the second year for College Fellow work has generally been praised by graduate students as a way for them to understand how the system works and how teaching works before they become full-fledged teachers. It's particularly helpful for people who did not do their undergraduate work in the US, insofar as it exposes them to undergraduate education in the US for the first time before they become responsible for conducting their own class.I believe some graduate students are unhappy with the extent of the teaching workload, and also with the formal requirement that an instructor can skip at most two classes in the quarter (and only one teaching class, and for that they must get another graduate student or faculty member to substitute for them). This is problematic for graduate students who want to attend conferences or give seminar talks (particularly in their job search year). I've heard that some graduate students skip more than that number of classes, thereby breaking the rules. (I've never skipped a class myself).At least in the major research areas, there is a critical mass of faculty and students making graduate student seminars and discussion groups effective. See http://math.uchicago.edu/calendar and http://math.uchicago.edu/graduate/gradseminars.shtml for an idea of the level of activity in various research areas.