Understanding the escape of Lyman continuum (LyC) and Lyman α (Lyα) photons from giant molecular clouds (GMCs) is crucial if we are to study the reionization of the Universe and to interpret spectra of observed galaxies at high redshift. To this end, we perform high-resolution, radiationmagneto-hydrodynamic simulations of GMCs with self-consistent star formation and stellar feedback. We find that a significant fraction (15-70%) of ionizing radiation escapes from the simulated GMCs with different masses (10 5 and 10 6 M), as the clouds are dispersed within about 2-5 Myr from the onset of star formation. The fraction of LyC photons leaked is larger when the GMCs are less massive, metal-poor, less turbulent, and less dense. The most efficient leakage of LyC radiation occurs when the total star formation efficiency of a GMC is about 20%. The escape of Lyα shows a trend similar to that of LyC photons, except that the fraction of Lyα photons escaping from the GMCs is larger (f Lyα ≈ f 0.27 900) and that a GMC with strong turbulence shows larger f Lyα. The simulated GMCs show a characteristic velocity separation of ∆v ≈ 120 km s −1 in the time-averaged emergent Lyα spectra, suggesting that Lyα could be useful to infer the kinematics of the interstellar and circumgalactic medium. We show that Lyα luminosities are a useful indicator of the LyC escape, provided the number of LyC photons can be deduced through stellar population modeling. Finally, we find that the correlations between the escape fractions of Lyα, ultraviolet photons at 1500 Å, and the Balmer α line are weak.