The SOLEDGE suite of codes has been specially designed to model the transition region from the hot core plasma to the first wall of tokamak, through the Last Closed Flux Surface (LCFS). It is designed to model electrostatic fluid turbulence for an isothermal plasma or for a plasma with temperature variations. Dedicated discretization algorithms have been implemented to handle equations for ion density, electron/ion temperatures and parallel momentum, both for the realistic cross-section of a diverted tokamak and for a three-dimensional cylindrical annulus. The efficient penalization method introduced in Ref.  has been implemented, allowing straightforward handling of solid obstacles by treating them as sink regions corresponding to strong plasma recombination in the solid state material. The SOLEDGE capability is exemplified here by simulating two equilibria: (i) a 3D cylindrical annulus and (ii) the cross-section of a diverted tokamak. In the annulus, the analysis of the impact of a secondary discrete limiter shows that the toroidal symmetry usually assumed for density and Mach profiles is broken. The density exhibits significant variations in the toroidal direction that extend over a large region of the scrape-off layer where magnetic field lines are connected to a secondary limiter. In the diverted geometry, computations show a transition from subsonic to supersonic flow in the vicinity of the X-point that is related to the location of particle sources and sinks between the edge connected region and the divertor region.