This paper presents the motivation, development, algorithms, and performance tests of some fundamental statistically based reconstruction algorithms in emission tomography (ET). The motivation is based on an analysis of the filtered backprojection method (FBP) of reconstruction as a statisticaf process in which it is shown that it corresponds to the wrong statistical distribution for the ET problem.The development of the target functions and algorithms for the maximum likelihood estimator method (MLE) and for a Bayesian method with entropy prior is shown in a manner consistent with the FBP analysis. The concept of a "feasible" image is discussed as a necessary condition for a reconstruction to be acceptable and the choice of a uniform image field as the initial guess for the iterative methods is justified in terms of the lack of prior information on the image to be reconstructed. Finally, a number of image and profile comparisons are presented in which the feasible images obtained by the above methods, and by the method of sieves, appear superior to FBP images. Real data from the ECAT-Ill tomograph are used for all the reconstructions.