Maximal functions are of great interest in Harmonic Analysis, PDEs, Ergodic theory, and other areas of Mathematics, dating back to the time of Hardy and Littlewood. In this talk, we shall focus primarily on the maximal wave propagator. A significant portion of this talk will be dedicated to their study on the Hardy spaces for Fourier integral operators. Our estimates on the Hardy spaces for Fourier integral operators will imply new and improved pointwise convergence results for the wave propagator. Based on a joint work with J. Rozendaal, N. Liu, and L. Song.