When does a knot in a 3-manifold have the smallest genus amongst all knots homologous to it? This is a basic but largely unexplored problem, and addressing it for lens spaces is closely connected with the Berge conjecture, which posits when surgery along a knot in the three-sphere can produce a lens space. I will describe this connection in greater detail and then exhibit some examples of non-simple genus-minimizing knots in lens spaces. This is joint work with Yi Ni.