A survey was developed to probe student understanding of quantum mechanics at the beginning of graduate instruction. The survey was administered to 202 physics graduate students enrolled in first-year quantum mechanics courses from seven universities at the beginning of the first semester. We also conducted one-on-one interviews with fifteen graduate or advanced undergraduate students who had just completed a course in which all the content on the survey was covered. Although students from some universities performed better on average than others, we found that students share universal difficulties understanding the concepts of quantum mechanics. The difficulties were often due to overgeneralizations of concepts learned in one context to other contexts where they are not directly applicable. Difficulties in distinguishing between closely related concepts and making sense of the formalism of quantum mechanics were common. The results of this study can sensitize instructors of first-year graduate quantum physics to some of the difficulties students are likely to face.