  A general continuous-state branching processes in random environment
(CBRE-process) is defined as the strong solution of a stochastic integral
equation. The environment is determined by a L\'evy process with no jump less
than $-1$. We give characterizations of the quenched and annealed transition
semigroups of the process in terms of a backward stochastic integral equation
driven by another L\'evy process determined by the environment. The process
hits zero with strictly positive probability if and only if its branching
mechanism satisfies Grey's condition. In that case, a characterization of the
extinction probability is given using a random differential equation with
singular terminal condition. The strong Feller property of the CBRE-process is
established by a coupling method. We also prove a necessary and sufficient
condition for the ergodicity of the subcricital CBRE process with immigration.
