Energy measurements play a fundamental role in the theory of quantum mechanics, yet there is evidence that the underlying concepts are difficult for many students, even after all undergraduate instruction. We present results from an investigation into student ability to determine the possible energies that can be measured for a given wave function and Hamiltonian, to determine the probabilities of each energy measurement and how they depend on time, and to recognize how a measurement of energy affects the state. By analyzing student responses to open-ended questions, we identify five broad, interrelated sets of conceptual and reasoning difficulties related to energy measurements. Data are drawn from sophomore-, junior-, and graduate-level quantum mechanics courses. Particular attention is paid to incorrect ideas that persist across all levels. [This paper is part of the Focused Collection on Upper Division Physics Courses.]