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A del Pezzo variety $X$ is a Fano variety whose anticanonical class has the form \[-K_X=(n-1)A,\] where $A$ is an ample line bundle and $n$ is the dimension of $X$. This is a higher dimensional analog notion of del Pezzo surfaces. I am going to discuss biregular and birational classifications of del Pezzo varieties admitting terminal singularities. The talk is based on a joint work with Alexander Kuznetsov (in preparation)