Abstract: MLEM (Maximum Likelihood-Expectation Maximization) has been widely used in Emission CT, however, in practice two draw backs are encountered: 1) due to the ill-conditioned nature of the reconstruction problem, reconstructions tent to take on an increasing non-smooth quality as the ML solution is approached; 2) convergence can be increasingly slow as the ML solution is approached. Maximum a posteriori (MAP) reconstruction has been shown to have significant advantages over MLEM in terms of noise reduction and the convergence speed. A review of development of MAP algorithms is presented in this paper.