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First we find effective bounds for the number of dominant rational maps $f:X\to Y$ between two fixed smooth projective varieties with ample canonical bundles. The bounds are of the type $\{A·K_X^n{\}}^{\{B·K_X^n{\}}^2}$, where $n=\dim X$, $K_X$ is the canonical bundle of $X$ and $A,B$ are some constants, depending only on $n$. Then we show that for any variety $X$ there exist numbers $c\left(X\right)$ and $C\left(X\right)$ with the following properties: For any threefold $Y$ of general type the number of dominant rational maps $f:X\to Y$ is bounded above by $c\left(X\right)$. The...