Given an $r$-uniform hypergraph $H$ and an $r$-uniform hypergraph $F$, define the relative Tur\'an number $ex(H,F)$ to be the maximum number of edges in an $F$-free subgraph of $H$. In this talk we discuss bounds for $ex(H,F)$ when $F$ is a loose cycle, Berge cycle, and a complete $r$-partite $r$-graph when the host $H$ is either arbitrary or when it is the random hypergraph $H_{n,p}^r$. \\ \\ This is joint work with Jiaxi Nie and Jacques Verstraete.