This paper presents a comprehensive synthesis of physics education research at the undergraduate level. It is based on work originally commissioned by the National Academies. Six topical areas are covered: (1) conceptual understanding, (2) problem solving, (3) curriculum and instruction, (4) assessment, (5) cognitive psychology, and (6) attitudes and beliefs about teaching and learning. Each topical section includes sample research questions, theoretical frameworks, common research methodologies, a summary of key findings, strengths and limitations of the research, and areas for future study. Supplemental material proposes promising future directions in physics education research.