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KaRMMa (Kappa Reconstruction for Mass MApping) performs curved-sky mass map reconstruction using a lognormal prior from weak-lensing surveys. It uses a fully Bayesian approach with a physically motivated lognormal prior to sample from the posterior distribution of convergence maps. The posterior distribution of KaRMMa maps are nearly unbiased in one-point and two-point functions and peak/void counts. KaRMMa successfully captures the non-Gaussian nature of the distribution of κ values in the simulated maps, and KaRMMa posteriors correctly characterize the uncertainty in summary statistics.