The conventional reconstruction techniques in positron emission tomography are based on deterministic principles, i.e., the stochastic nature of the measurements is completely ignored. This approach is quite acceptable in situations where the number of detector counts is sufficiently high. In practice, however, there are limitations to both the amount of activity administered to the patient, and the acquisition time, which usually results in a relatively low number of counts. It is known that the statistical fluctuations of these measurements can give rise to severe reconstruction artefacts. Stochastic methods are preferable in this context, as they use the information contained in the measurements more efficiently. In this paper, we present a modified version of Rockmore and Macovski's maximum-likelihood formula (for Poission measurement statistics), which takes into account the attenuation of the subject. Our implementation, based on an iterative algorithm, produces excellent images with low noise content and virtually no artefacts. Furthermore, the resolution comes close to the theoretical limit imposed by the sampling distance. A suboptimal version of the algorithm is also presented. Comparisons are made between the error performance of both algorithms and that of the filtered backprojection method. In order to make a fair comparison, we include two curves (corresponding to different convolution filters) for the conventional method. The absence of artefacts and lower noise, especially in low intensity areas of the images, are distinct advantages for diagnostic purposes. Alternatively, acceptable recon-structions can be obtained with shorter acquisition times or lower radionuclide doses for the patient.