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Mapper is a popular Toplological Data Analysis tool that is used to investigate the shape and features of point cloud data. Other than the data, the Mapper algorithm needs few more parameters. It needs a topological space Y, an open cover of Y, and a function (called a filter function) that maps your data into Y. In context of the application, sometimes there is a natural choice for the filter function, and hence Y. However, it is rarely the case that there is a natural choice for the open cover of Y. In this talk, we will investigate recent work concerning how one can optimally choose the open cover for Y, some related inverse Mapper problems, its Reeb space versions, and if we have time, its similarities with well-known problems concerning intersection patterns of convex hulls.