I will start with a gentle introduction on Poisson boundaries for random walks on groups, and give two basic examples. Then I will consider a class of groups on two generators and one relation, which play an important role in geometric group theory. The groups under consideration are called Baumslag-Solitar groups. I will consider random walks driven by a probability measure with finite first moment on such groups, and study the behaviour at infinity on them: convergence to the boundary and the Poisson boundary.