  We consider families ${\cal F}(\Delta)$ consisting of complex
$(n-1)$-dimensional projective algebraic compactifications of $\Delta$-regular
affine hypersurfaces $Z_f$ defined by Laurent polynomials $f$ with a fixed
$n$-dimensional Newton polyhedron $\Delta$ in $n$-dimensional algebraic torus
${\bf T} =({\bf C}^*)^n$. If the family ${\cal F}(\Delta)$ defined by a Newton
polyhedron $\Delta$ consists of $(n-1)$-dimensional Calabi-Yau varieties, then
the dual, or polar, polyhedron $\Delta^*$ in the dual space defines another
family ${\cal F}(\Delta^*)$ of Calabi-Yau varieties, so that we obtain the
remarkable duality between two {\em different families} of Calabi-Yau
varieties. It is shown that the properties of this duality coincide with the
properties of {\em Mirror Symmetry} discovered by physicists for Calabi-Yau
$3$-folds. Our method allows to construct many new examples of Calabi-Yau
$3$-folds and new candidats for their mirrors which were previously unknown for
physicists. We conjecture that there exists an isomorphism between two
conformal field theories corresponding to Calabi-Yau varieties from two
families ${\cal F}(\Delta)$ and ${\cal F}(\Delta^*)$.
