Muon-spin rotation (muSR) studies of the oxygen isotope (^{16}O/^{18}O) effect (OIE) on the in-plane magnetic field penetration depth lambda_{ab} in cuprate high-temperature superconductors (HTS) are presented. First, the doping dependence of the OIE on the transition temperature T_c in various HTS is briefly discussed. It is observed that different cuprate families show a similar doping dependence of the OIE on T_c. Then, bulk muSR, low-energy muSR, and magnetization studies of the total and site-selective OIE on lambda_{ab} are described in some detail. A substantial OIE on lambda_{ab} was observed in various cuprate families at all doping levels, suggesting that cuprate HTS are non-adiabatic superconductors. The experiments clearly demonstrate that the total OIE on T_c and lambda_{ab} arise from the oxygen sites within the superconducting CuO_2 planes, demonstrating that the phonon modes involving the movement of planar oxygen are dominantly coupled to the supercarriers. Finally, it is shown that the OIE on T_c and lambda_{ab} exhibit a relation that appears to be generic for different families of cuprate HTS. The observation of these unusual isotope effects implies that lattice effects play an essential role in cuprate HTS and have to be considered in any realistic model of high-temperature superconductivity.
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