The Virasoro-Shapiro (VS) amplitude describes closed string scattering at tree level in string theory, and is the most basic quantity in string theory. String theory is best understood in AdS background, where AdS/CFT gives a non-perturbative definition. But surprisingly the corresponding amplitude in AdS was not computed until recently, as worldsheet calculations are hard in the presence of Ramond-Ramond flux. We review the calculation of the AdS VS for AdS_5 x S^5 in an expansion around flat space, which follows from an ansatz for the amplitude in terms of a worldsheet integral of a special class of functions, combined with the block expansion of the flat space limit of the correlator in the dual CFT. We then generalize this calculation to AdS_4 x CP^3 and AdS_3 x S^3 x K3 and AdS_3 x S^3 x T^4. Our results give predictions for CFT data of heavy single trace operators, e.g. the Konishi, that can be compared to integrability.