In this paper general simulations of algorithms designed for fully operational BSP machines on BSP machines with fault y or unavailable processors, are deveioped. The fail-stop model is considered for the fault occurrences, that is, if a processor fails or becomes unavailable, it remains so until the end of the computation. The faults are random, in the sense that a processor may fail independently with probability at most a, where a is a constant. Two possible settings for fault occurrences are considered: the static case where the faults are static (the faulty or unavailable processors are already known at the beginning of the computation), and the dynamic case where the processors may become faulty or unavailable during the computation. In the case of static faults, a simulation of an n-processor fault-free BSP machine on a faulty n-processor BSP machine is presented with constant slowdown per local computation step and O(log n . max{ L, g }) slowdown per communication step, given that a preprocessing has been done that needs O(n/ log n max{L, g}) time (L and g are the parameters of the simulating BSP machine). In the case of dynamic faults, a simulation of an n-processor fault-free BSP machine on an cn log nprocessor faulty BSP machine is presented. No dynamic faults may occur during certain periods of the simulation. The simulations are randomized and Monte Carlo: they are guaranteed to be correct with high probability, and the time bounds always hold. To our knowledge, no previous work on the fault tolerance of the BSP model exists.