In this talk, we will introduce a class of oscillatory integral operators with the kernel being smooth function and compact support. Stein and Phong systematacially investigated those operators and obtained the sharp $L^2$ decay estimates.In fact, Stein's results answered the important conjecture which was put by the distinguished mathematician Arnold. That is, the sharp decay estimate is determinated by the Newton polyhedron of the phase function of the oscillatory integral. Finally, we give the sharp $L^p$ decay estimates of the oscillatory integral operators with homogeneous polynomial phases. As a consequence, we also give sharp $L^p$-boundedness of the generalized Fourier transform. E.M.Stein attributed the oscillatory integral operator to one of the most important three operators in Harmonic Analysis. Actually, the Fourier transform and Bochner-Riesz means are two classical oscillatory integral operators.