Geometrostatics is the geometric theory of static isolated relativistic systems modeling for example static stars or black holes in our universe. It is governed by a system of (mainly) elliptic PDEs relating a $3$-dimensional Riemannian metric and a positive function describing the geometry of space at any given point of time and the passage of time, respectively. While these PDEs have been intensively studied in the last century from both analytic and physical viewpoints, their geometric nature had not been exploited to its full extent. We will present a number of new insights into geometrostatics that mainly rely on a geometric approach but are also interrelated with analytic and physical aspects of the theory. For example, we will discuss geodesics of the corresponding Lorentzian spacetimes, uniqueness questions, mass and center of mass of geometrostatic systems, and last but not least their Newtonian limit -- the question of whether and how these relativistic systems have Newtonian counterparts.