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Turán problems ask for the maximum number $ex(n,F)$ of edges that an $n$-vertex graph $H$ can have without containing a copy of the forbidden graph $F$. These problems are the starting points of extremal graph theory and there have been an enormous amount of research in the area in the past century. There exist many generalizations and variants to this kind of problems. In my talk, I will survey some recently introduced notions and the first couple of results concerning these notions all of which involve the degrees of either all vertices of the graph $H$ or of all vertices of the copy of $F$ in $H$.

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