© 2018 Vieten et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. The science of meditation has grown tremendously in the last two decades. Most studies have focused on evaluating the clinical effectiveness of mindfulness-based interventions, neural and other physiological correlates of meditation, and individual cognitive and emotional aspects of meditation. Far less research has been conducted on more challenging domains to measure, such as group and relational, transpersonal and mystical, and difficult aspects of meditation; anomalous or extraordinary phenomena related to meditation; and post-conventional stages of development associated with meditation. However, these components of meditation may be crucial to people's psychological and spiritual development, could represent important mediators and/or mechanisms by which meditation confers benefits, and could themselves be important outcomes of meditation practices. In addition, since large numbers of novices are being introduced to meditation, it is helpful to investigate experiences they may encounter that are not well understood. Over the last four years, a task force of meditation researchers and teachers met regularly to develop recommendations for expanding the current meditation research field to include these important yet often neglected topics. These meetings led to a cross-sectional online survey to investigate the prevalence of a wide range of experiences in 1120 meditators. Results show that the majority of respondents report having had many of these anomalous and extraordinary experiences. While some of the topics are potentially controversial, they can be subjected to rigorous scientific investigation. These arenas represent largely uncharted scientific terrain and provide excellent opportunities for both new and experienced researchers. We provide suggestions for future directions, with accompanying online materials to encourage such research.