This article reviews numerical experiments on the three-body problem carried out at the Leningrad University Astronomical Observatory during the past 20 years. Systematic studies of triple systems with negative total energy have yielded the following main results. Most (≈95%) of the systems decay; the decay always occurs after a close triple approach of the components. In a system with unequal masses, the escaping body usually has the smallest mass. A small fraction (≈5%) of quasi-stable systems is formed if the angular momentum is non-zero. The qualitative evolution in three-dimensional cases is the same as for planar systems. Small changes in initial conditions sometimes lead to substantial differences in the final outcome. The decay of triple systems is a stochastic process similar to radioactive decay. The estimated mean lifetime is ≈100 crossing times for equal-mass components and decreases for increasing mass dispersion. A classification of the close triple approaches which lead to immediate escape is given for equal-mass systems as well as for selected sets of unequal components. Detailed studies of close triple approaches by computer simulations reveal that the early evolutions is determined by the initial ratio of the interaction forces. The review concludes by discussing applications of the results to observational problems of stellar and extragalactic systems.