  First we find effective bounds for the number of dominant rational maps $f:X
\rightarrow Y$ between two fixed smooth projective varieties with ample
canonical bundles. The bounds are of the type $\{A \cdot K_X^n\}^{\{B \cdot
K_X^n\}^2}$, where $n=dimX$, $K_X$ 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 $c(X)$ and $C(X)$ with the following properties: For
any threefold $Y$ of general type the number of dominant rational maps $f:X \r
Y$ is bounded above by $c(X)$. The number of threefolds $Y$, modulo birational
equivalence, for which there exist dominant rational maps $f:X \r Y$, is
bounded above by $C(X)$. If, moreover, $X$ is a threefold of general type, we
prove that $c(X)$ and $C(X)$ only depend on the index $r_{X_c}$ of the
canonical model $X_c$ of $X$ and on $K_{X_c}^3$.
