Inherent in the practice of apprentice-model undergraduate research (UR) is a fundamental tension between the educational goals of UR and its basis in faculty scholarship. This tension leads to challenges for faculty in guiding student researchers in their daily work and in positioning their own UR work within institutionally bifurcated domains of teaching and research. It also generates a disconnect when it comes to measuring the outcomes of UR. Traditional outcome measures emphasize students' career outcomes and research productivity, while education research has documented students' personal and professional learning from UR, including new skills and understandings of disciplinary inquiry, growth in confidence and responsibility, and scientific identity development. Thus far, self-report measures including surveys and interviews have dominated this young body of research. I discuss why assessing the outcomes of apprentice-model undergraduate research is inherently difficult, outline some strengths and limitations of the approaches tried to date, and suggest areas for future research, including the design and measurement challenges that arise in attempting to incorporate undergraduate research into courses.