The worldwide elimination of measles and rubella is feasible, but not without overcoming the substantial challenge of vaccine hesitancy. This challenge is complicated by the spread of misinformation and disinformation fueled by rapidly progressing technologies and evolving forms of online communication. The recent COVID-19 pandemic has only added further complexity to this challenge. However, considerable progress has been made in understanding the scope of the problem and the complex factors that influence vaccine hesitancy. Our understanding of evidence-based strategies for addressing vaccine hesitancy has grown significantly, including evidence for effective communication and behavioral interventions. In this article, we review measles and rubella vaccines and vaccine hesitancy. We then provide an overview of evidence-based strategies for addressing vaccine hesitancy, including communication strategies and behavioral interventions. This article is relevant to healthcare professionals, health system leaders, public health professionals, policymakers, community leaders, and any individuals who have a role in addressing vaccine hesitancy in their communities. Finally, we review future directions and major areas of research need.