Fosite implements a method for the solution of hyperbolic conservation laws in curvilinear orthogonal coordinates. It is written in Fortran 90/95 integrating object-oriented (OO) design patterns, incorporating the flexibility of OO-programming into Fortran 90/95 while preserving the efficiency of the numerical computation. Although mainly intended for CFD simulations, Fosite's modular design allows its application to other advection problems as well. Unlike other two-dimensional implementations of finite volume methods, it accounts for local conservation of specific angular momentum. This feature turns the program into a perfect tool for astrophysical simulations where angular momentum transport is crucial. Angular momentum transport is not only implemented for standard coordinate systems with rotational symmetry (i.e. cylindrical, spherical) but also for a general set of orthogonal coordinate systems allowing the use of exotic curvilinear meshes (e.g. oblate-spheroidal). As in the case of the advection problem, this part of the software is also kept modular, therefore new geometries may be incorporated into the framework in a straightforward manner.