Networked infrastructures operated under highly loaded conditions are vulnerable to catastrophic cascading failures. For example, electric power transmission systems must be designed and operated to reduce the risk of widespread blackouts caused by cascading failure. There is a need for analytically tractable models to understand and quantify the risks of cascading failure. We study a probabilistic model of loading dependent cascading failure by approximating the propagation of failures as a Poisson branching process. This leads to a criticality condition for the failure propagation. At criticality there are power tails in the probability distribution of cascade sizes and consequently considerable risks of widespread catastrophic failure. Avoiding criticality or supercriticality is a key approach to reduce this risk. This approach of minimizing the propagation of failure after the cascade has started is complementary to the usual approach of minimizing the risk of the .rst few cascading failures. The analysis introduces a saturating form of the generalized Poisson distribution so that supercritical systems with a high probability of total failure can be considered.