A Monte Carlo code is used to calculate the fast fission factor epsilon and the fast fission ratio delta in both homogeneous and heterogeneous systems. The following results are obtained: for an infinite block of natural uranium, delta = 0.39, epsilon = 1.247; for an infinite block of beryllium, epsilon = 1.078; for beryllium oxide, = 1.046. In addition, results are given for homogeneous mixtures of uranium and beryllium. Calculations are also carried out for uranium-water lattices and compared with experimental results for slightly enriched uranium rods and slabs, and uranium oxide rods. Other results show the increase in epsilon when uranium fuel elements are surrounded by beryllium cladding of varying thickness. (auth)