To help faculty use research-based materials in a more significant way, we learn about their perceived needs and desires and use this information to suggest ways for the physics education research community to address these needs. When research-based resources are well aligned with the perceived needs of faculty, faculty members will more readily take them up. We used phenomenographic interviews of ordinary physics faculty and department chairs to identify four families of issues that faculty have around research-based assessments (RBAs). First, many faculty are interested in using RBAs but need help with the practicalities of administering RBAs: how to find them, which ones there are, and how to administer them. Second, at the same time, many faculty think that RBAs are limited and do not measure many of the things they care about, or are not applicable in their classes. They want assessments to measure skills, perceptions, and specific concepts. Third, many faculty want to turn to communities of other faculty and experts to help them interpret their assessment results and suggest other ways to do assessment. They want to better understand their assessment results by comparing to others and interacting with faculty from other schools to learn about how they do assessment. Fourth, many faculty consider their courses in the broader contexts of accountability and their departments. They want help with assessment in these broader contexts. We also discuss how a faculty member's role in their department and type of institution influence their perceived wants and needs around assessment. [This paper is part of the Focused Collection on Preparing and Supporting University Physics Educators.]