Since the publication of Shepp and Vadi's [ 14] maximum likelihood reconstruction algorithm for emission tomography (ET), many medical research centers engaged in ET have made an effort to change their reconstruction algorithms to this new approach. Some have succeeded, while others claim they could not adopt this new approach primarily because of limited computing power. In this paper, we discuss techniques for reducing the computational requirements of the reconstruction algorithm. Specifically, the paper discusses the data structures one might use and ways of taking advantage of the geometry of the physical system. The paper also treats some of the numerical aspects of the EM (expectation maximization) algorithm, and ways of speeding up the numerical algorithm using some of the traditional techniques of numerical analysis.