This paper summarizes the results obtained by the team Heliosheath Processes and the Structure of the Heliopause: Modeling Energetic Particles, Cosmic Rays, and Magnetic Fields supported by the International Space Science Institute in Bern, Switzerland. We focus on the physical processes occurring in the outer heliosphere, especially at its boundary called the heliopause (HP), and in the LISM. The importance of magnetic field, charge exchange between atoms and ions, and solar cycle on the heliopause topology and observed heliocentric distances to different heliospheric discontinuities are discussed. It is shown that time-dependent boundary conditions are necessary to describe the heliospheric asymmetries detected by the Voyager spacecraft. We also discuss the structure of the HP, especially due to its instability and magnetic reconnection. It is demonstrated that the Rayleigh-Taylor instability of the nose of the HP creates consecutive layers of the interstellar and heliospheric plasma which are magnetically connected to different sources. This may be a possible explanation of abrupt changes in the galactic and anomalous cosmic ray fluxes observed by Voyager 1 when it was crossing the HP structure for a period of about one month in the summer of 2012. This paper also discusses the plausibility of fitting simulation results to a number of observational data sets obtained by in situ and remote measurements. The distribution of magnetic field in the vicinity of the HP is discussed in the context of Voyager measurements. We discuss the transport of energetic particles in the inner and outer heliosheath, concentrating on the anisotropic spatial diffusion diffusion tensor and the pitch-angle dependence of perpendicular diffusion and demonstrate that the latter can explain the observed pitch-angle anisotropies of both the anomalous and galactic cosmic rays in the outer heliosheath.
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