The paper under review deals with oscillatory integrals and their relationship with the local and global smoothing properties of dispersive equations of the type \[ \partial_ tu-iP(D)u=0,\quad x\in\mathbb{R}^ n,\;\;t\in\mathbb{R}, \] where \(D=(1/i)(\partial_ 1,\ldots,\partial_ n)\), \(\partial_ j=\partial/\partial x_ j\), and \(P(D)\) is defined via its real symbol, i.e., \[ P(D)f(x)=\int\exp(ix\xi)P(\xi)\hat f(\xi) d\xi. \] The authors study also some applications of these smoothing properties to nonlinear problems and their link with restriction theorem for the Fourier transform and positive convergence results.