This talk will highlight some interesting maps of topological spaces giving rise to maps of face posets and how an interplay of combinatorics and topology may help reveal structure of both the image and the fibers of the map. Much of the talk will focus on joint work with Alex Engstrom and Bernd Sturmfels regarding images of monomial maps on probability spaces, namely maps given by d monomials in n variables where these variables are probabilities, so maps from an n-dimensional unit cube to a d-dimensional unit cube. We dub the images of such maps as 'toric cubes'. The motivating example is the edge product space of phylogenetic trees. For any monomial map, we provide a cell structure for the image which in fact is a regular CW decomposition of the image; this involves introducing a new family of maps which are generically injective. As time permits, we will briefly highlight other interesting examples of generically injective maps coming from total positivity theory and from electrical networks as well as some new combinatorial-topological tools for studying homeomorphism type and fibers of such maps.