ABSTRACT. Dykstra’s cyclic projections algorithm allows one to compute best approximations to any point x in a Hilbert space from the intersection C = ∩_{1}^{r}C_{i} of a finite number of closed convex sets C_{i}, by reducing it to a sequence of best approximation problems from the individual sets C_{i}. Here we present two generalizations of this algorithm. First we allow the number of sets C_{i} to be infinite rather than finite; secondly, we allow a random, rather than cyclic, ordering of the sets C_{i}.