Fully three-dimensional positron emission tomography is considered and a reconstruction algorithm derived. The reconstruction problem is formulated mathematically as a three-dimensional convolution integral of a point spread function with an unknown positron activity distribution and is solved by Fourier transform methods. Performance of the algorithm is evaluated using both simulated phantom data produced by a Monte Carlo computer program and phantom data obtained from the University of Chicago/Searle Positron Camera. It is concluded that the method is computationally feasible and results in accurate reconstructions.