  We show that the existence of a universal structure, for an approximable
class $\mathcal{C}$ of countable structures, implies the existence of a generic
structure for $\mathcal{C}$. We also show that the converse is not true. As a
consequence, we provide several new examples of weak Fra\"iss\'e classes of
finite graphs. Finally, we show that the class of all countable $C_4$-free
graphs does not contain a generic structure, strengthening a result of A.
Hajnal and J. Pach.
