In Extremal Graph Theory one is interested in relations between various graph invariants. Given a property $P$ and an invariant $u$ for a family $\mathcal{F}$ of graphs, we wish to determine the maximum value of $u(G)$ among all graphs $G$ in $\mathcal{F}$ satisfying the property $P$. The optimal value $u(G)$ is called the {\it extremal number} and the graphs attaining this value are called {\it extremal graphs}. A principal example of such an extremal problem is the so-called Turan type problem, initiated by Hungarian mathematicians Turan and Erd\H{o}s in 1940s. In this talk, we will discuss recent results on several prominent conjectures of Erdős-Simonovits on extremal graphs.