It has now been twenty years since the publication of W. T. Ingram and W. S. Mahavier’s landmark paper, “Inverse limits of upper semi-continuous set valued functions” (Houston Journal of Mathematics, 2006, vol. 32, no. 1, p. 119-130). For all these years, generalized inverse limits whose factor spaces are arcs have been studied intensively by researchers around the world. However, generalized inverse limits whose factor spaces are circles have been far less thoroughly studied, and could offer researchers a whole new frontier to explore. We state some questions about these spaces and provide various examples, including an example of a generalized inverse limit on circles (with a single, continuum-valued bonding function) that gives rise to an indecomposable continuum.