Recently, a very huge amount of Jpsi sample, 225 million events taken in 2009 and 1087 million events taken in 2012, has been accumulated at BESIII. The data set offers a unique opportunity to study the baryon decay processes and search for the rare decays of eta and eta’ mesons. Based on the effective Lagrangian approach, we studied the Jpsi->ppbar phi decay in an isobar resonance model with the assumption that the phi-meson is produced from intermediate nucleon resonances. Using the 2009 data, we present results of a study of the decay Jpsi->gamma Lambda Lambdabar, and report the first observation of isospin violating decay Jpsi->Lambda Sigmabar^{0}+c.c.. Based on the total 1.3 billion data, we study the rare electromagnetic decays: eta, eta’-> pi^{0} gamma gamma in processes of Jpsi-> gamma eta and Jpsi->gamma eta’, respectively. The eta’->pi^{0} gamma gamma decay is observed for the first time.