  Let $X$ be a normal projective variety admitting a polarized or int-amplified
endomorphism $f$. We list up characteristic properties of such an endomorphism
and classify such a variety from the aspects of its singularity, anti-canonical
divisor and Kodaira dimension. Then we run the equivariant minimal model
program with respect to not just the single $f$ but also the monoid $SEnd(X)$
of all surjective endomorphisms of $X$, up to finite-index. Several
applications are given. We also give both algebraic and geometric
characterizations of toric varieties via polarized endomorphisms.
