Abstract: The Plancherel Theorem states that if $f$ is an $L^2$ function , then $\hat f$, the Fourier transform of the function $f$, is also in $L^2$. Moreover, the map which takes $f$ to $\hat f$ is a bounded linear operator. \v The boundedness of the Fourier transform rests on the fact that the phase function $S(x,y) = x_1y_1+\cdots x_ny_n$ is non-degenerate (its Hessian determinant is non-vanishing). The case of degenerate phases is more complicated, and has been the focus of much work in harmonic analysis. \v In our talk we shall describe some joint work with D.H. Phong and E.M. Stein where we study the case where the phase $S$ is a (possibly degenerate) polynomial : We shall show that the corresponding operator is bounded in $L^p$ with bounds which are (sharply) determined by the Newton Polygon of $f$. The bounds are stable in the sense they are locally independent of $S$: Thus they continue to hold under small perturbations of the coefficients of the polynomial $S$.