Given a graph , there are several natural hypergraph families one can define. Among the least restrictive is the family of so-called Berge copies of the graph . In this talk, we discuss Turán problems for families in -uniform hypergraphs for various graphs . In particular, we are interested in general results in two settings: the case when is large and is any graph where this Turán number is shown to be eventually subquadratic, as well as the case when is a tree where several exact results can be obtained. The results in the first part are joint with Grósz and Methuku, and the second part with Győri, Salia and Zamora.