We consider continuous state branching processes (CSBP's for short) with additional multiplicative jumps, which we call catastrophes. Informally speaking, the dynamics of the CSBP is perturbed by independent random catastrophes which cause negative (or positive) jumps to the original process. These jumps are described by a Lévy process with paths of bounded variation. Conditionally on these jumps, the process still enjoys the branching property. We construct this class of processes as the unique solution of a SDE and characterize their Laplace exponent as the solution of a backward ODE. We can then study their asymptotic behavior and establish whether the process becomes extinct. For a class of processes for which extinction and absorption coincide (including the alpha-stable CSBP plus a drift), we determine the speed of extinction of the process. Then, three subcritical regimes appear, as in the case for branching processes in random environments. To prove this, we study the asymptotic behavior of a certain divergent exponential functional of Lévy processes. Finally, we apply these results to a cell infection model, which was a motivation for considering such CSBP's with catastrophes.