Different notions of boundaries are useful when studying topological groups. In the case of locally compact simply connected groups, the Poisson (or Furstenberg-Poisson) boundary can be thought of as the objects appearing in the limit of random walks on a group. I'll begin with discussing random walks on groups. Then, I will cover the definitions and motivation for the boundary in the case of discrete/countable groups, proving some theorems relating tail events to harmonic functions along the way.