Let $(X,\rho )$, $(Y,\sigma )$ be metric spaces and $f:X\to Y$ an injective mapping. We put ${\parallel f\parallel }_{Lip}=sup\{\sigma (f\left(x\right),f\left(y\right))/\rho (x,y)$; $x,y\in X$, $x\ne y\}$, and $dist\left(f\right)={\parallel f\parallel }_{Lip}.{\parallel f^{-1}\parallel }_{Lip}$ (the distortion of the mapping $f$). Some Ramsey-type questions for mappings of finite metric spaces with bounded distortion are studied; e.g., the following theorem is proved: Let $X$ be a finite metric space, and let $\epsilon >0$, $K$ be given numbers. Then there exists a finite metric space $Y$, such that for every mapping $f:Y\to Z$ ($Z$ arbitrary metric space) with $dist\left(f\right)<K$ one can find a mapping $g:X\to Y$, such that both the mappings $g$ and ${f|}_{g\left(X\right)}$ have distortion at...