  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.
