Well-designed interactions for visualization systems hold great promise both for empowering people to effectively complete their tasks and for providing more natural and engaging user experiences. Leveraging advancements in hardware and software technology, the visualization research community has made considerable progress providing novel input and interaction experiences. These technologies continue to advance at a fast pace, and thus it seems timely to look back at what has been achieved so far and contemplate what might be possible in the future. In this monograph, we first present a condensed summary of research efforts investigating post-WIMP interaction techniques in visualization systems. We also include research from the broader HCI community and several product releases from industry that we believe to be relevant and have influenced visualization interfaces. Furthermore, we reflect on our own projects that investigated post-WIMP InfoVis interaction and systems. We discuss the main challenges we faced and lessons we learned, and we reflect on how our perspectives and viewpoints on post-WIMP for InfoVis have evolved over the course of these projects. Finally, we identify several open research directions that will help realize the full potential of post-WIMP interaction for and with InfoVis, expanding the boundaries of InfoVis and reaching a broader audience.