For a proper understanding of the fundamental physical properties of compact objects their environment can not be neglected. The matter and energy distribution in the vicinity of compact objects is an important source for astrophysical observations, like X–rays emitted by accretion disks. Although the basic concept of accretion is clear several important aspects are still not completely understood. Often numerical simulations of high complexity, like general relativistic magnetohydrodynamical (GRMHD) codes, are utilized to describe accretion flows. However, to extend our fundamental physical understanding of the accretion process analytical models have been and will be the most important tool. Furthermore, it is still a serious problem of computational GRMHD that only very few exact analytical or semi-analytical solutions exist that can be used to test the numerical simulations. To address these problems, we will focus on analytical models of accreting matter. We will evolve known analytical models to include new aspects, develop new analytical models to cover setting not studied before in such a context, apply the models to generalized compact objects, and consider alternative and modified gravity theories. The analytical results will also serve as test beds for numerical codes and to predict observable consequences.