Many problems that arise in mathematics, science, and engineering can be formulated as solving a system of parameterized polynomial equations, which depend on data from the application. Some examples of particular interest are scene reconstruction in computer vision, motion of mechanisms in kinematics, and oscillations, bistability, and other dynamical properties in chemical reaction networks. The parameters in these cases are image data points, leg lengths, and reaction rate constants, respectively. Characteristics such as the number of real solutions or the behavior/structure of the real solutions change depending on the values of these parameters. This talk will discuss projects within these applications that seek to describe the geography of the parameter space connected to these characteristics. It will also highlight next steps in being able to apply these techniques to visualizing and exploring alternate, non-standard real structures, such as the toric varieties related to the Schrödinger operator on the hexagonal lattice underlying graphene.