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Crossing of the heliopause by Voyager 1 was accompanied by a number of surprises, e.g., the sequence of increases and decreases in the galactic cosmic ray (GCR) flux, magnetic field behavior, flow direction, unexpectedly small heliocentric distance of the heliopause, nearly zero gradient in the GCR flux beyond the heliopause, etc. Numerical simulations of the heliospheric interface revealed instability of the heliopause caused initially by charge exchange between ions and neutral atoms and further enhanced by the shear flow. Magnetic field distribution in the inner heliosheath reveals signatures of substantial dissipation immediately beyond the termination shock. We present the results of numerical simulations which show that the heliopause instability may be possible explanation of the above-mentioned observations. In particular, the heliopause instability creates multiple layers of the solar wind (SW) and local interstellar medium (LISM) plasma following each other. We show that magnetic-line connection of these layers creates conditions for the GCR flux variations similar to Voyager observations, provided that the parallel diffusion dominates over the perpendicular diffusion in the LISM magnetic field. We also analyze magnetic reconnection in the turbulent plasma near the heliopause. The analysis is performed using both local, in-the-box simulations and in the global context of the SW-LISM interaction. We show that magnetic reconnections are typical for many regions near the heliopause. However, it is more likely for Voyagers to see only the global consequences of reconnections that occur at some distance from the crossing points. Using full-particle and hybrid simulations, we also analyze a number of issues related to the pickup ion behavior beyond the heliopause and, in particular, the validity of the scenario in which the IBEX ribbon a line-of-sight selection effect based on the idea that the energetic neutral atoms detectors should preferentially observe the flux perpendicular to the interstellar magnetic field lines draping around the heliopause, as in Heerikhuisen et al. (2010).