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Large-scale disturbances generated by the Sun’s dynamics first propagate through the heliosphere, influence the heliosphere’s outer boundaries, and then traverse and modify the very local interstellar medium (VLISM). The existence of shocks in the VLISM was initially suggested by Voyager observations of the 2-3 kHz radio emissions in the heliosphere. A couple of decades later, both Voyagers crossed the definitive edge of our heliosphere and became the first ever spacecraft to sample interstellar space. Since Voyager 1’s entrance into the VLISM, it sampled electron plasma oscillation events that indirectly measure the medium’s density, increasing as it moves further away from the heliopause. Some of the observed electron oscillation events in the VLISM were associated with the local heliospheric shock waves. The observed VLISM shocks were very different than heliospheric shocks. They were very weak and broad, and the usual dissipation via wave-particle interactions could not explain their structure. Estimates of the dissipation associated with the collisionality show that collisions can determine the VLISM shock structure. According to theory and models, the existence of a bow shock or wave in front of our heliosphere is still an open question as there are no direct observations yet. This paper reviews the outstanding observations recently made by the Voyager 1 and 2 spacecraft, and our current understanding of the properties of shocks/waves in the VLISM. We present some of the most exciting open questions related to the VLISM and shock waves that should be addressed in the future.