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Abstract: Let G_{n,p} denote the random n-vertex graph obtained by including each edge independently and with probability p.  Given a graph F, let ex(G_{n,p},F) denote the size of a largest F-free subgraph of G_{n,p}.  When F is non-bipartite, the asymptotic behavior of ex(G_{n,p},F) was determined in breakthrough work independently by Conlon-Gowers and by Schacht.  Much less is known when F is bipartite.  In this talk we will survey the known bounds on ex(G_{n,p},F) when F is bipartite, as well as the general techniques used for these problems.  We will also discuss the random Turán problem for r-partite r-uniform hypergraphs as time permits.