We present a high-statistics measurement of differential cross sections and the total cross section for the process gamma gamma -> pi^0 pi^0 in the kinematic range 0.6 GeV <= W <= 4.0 GeV and |cos theta*| <= 0.8, where W and theta* are the energy and pion scattering angle, respectively, in the gamma gamma center-of-mass system. Differential cross sections are fitted to obtain information on S, D_0, D_2, G_0 and G_2 waves. The G waves are important above W ~= 1.6 GeV. General behavior of partial waves is studied by fitting differential cross sections in a simple parameterization where amplitudes contain resonant contributions and smooth background. The D_2 wave is dominated by the f_2(1270) meson whose parameters are consistent with the with the current world averages. The D_0 wave contains a f_2(1270) component, whose fraction is fitted. For the S wave, the f_0(980) parameters are found to be consistent with the values determined from our recent pi^+ pi^- data. In addition to the f_0(980), the S wave prefers to have another resonance-like contribution whose parameters are obtained.
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