Economic projections point to a need for approximately 1 million more STEM professionals than the U.S. will produce at the current rate over the next decade if the country is to retain its historical preeminence in science and technology. To meet this goal, the United States will need to increase the number of students who receive undergraduate STEM degrees by about 34% annually over current rates. The title of this report, "Engage to Excel," applies to students, faculty, and leaders in academia, industry, and government. Students must be engaged to excel in STEM fields. To excel as teachers, faculty must engage in methods of teaching grounded in research about why students excel and persist in college. Moreover, success depends on the engagement by great leadership. Leaders, including the President of the United States; college, university and business leadership; and others, must encourage and support the creation of well-aligned incentives for transforming and sustaining STEM learning. They also must encourage and support the establishment of broad-based reliable metrics to measure outcomes in an ongoing cycle of improvement. Transforming STEM education in U.S. colleges and universities is a daunting challenge. The key barriers involve faculty awareness and performance, reward and incentive systems, and traditions in higher education. The recommendations in this report address the most significant barriers and use both tangible resources and persuasion to inspire and catalyze change. Attacking the issue from numerous angles and with various tools is aimed at reaching a point at which the movement will take on a momentum of its own and produce sweeping change that is sustainable without further Federal intervention. The President's Council of Advisors on Science and Technology (PCAST) proposes five overarching recommendations to transform undergraduate STEM education during the transition from high school to college and during the first two years of undergraduate STEM education: (1) Catalyze widespread adoption of empirically validated teaching practices; (2) Advocate and provide support for replacing standard laboratory courses with discovery-based research courses; (3) Launch a national experiment in postsecondary mathematics education to address the math preparation gap; (4) Encourage partnerships among stakeholders to diversify pathways to STEM careers; and (5) Create a Presidential Council on STEM Education with leadership from the academic and business communities to provide strategic leadership for transformative and sustainable change in STEM undergraduate education. Appended are: (1) Experts Providing Input to PCAST; (2) Acknowledgements; (3) STEM Higher Education Enrollment, Persistence, and Completion Data; (4) Economic Analysis of STEM Workforce Need; (5) Evidence of the Mathematics Preparation Gap; (6) Efficacy of Various Classroom Methods; (7) Review of Evidence that Research Experiences have Impacts on Retention; (8) Effective Programs to Improve STEM Undergraduate Education; and (9) References for Tables 2, 3, and 4. (Contains 17 tables, 21 figures, 11 boxes, 151 footnotes.)