  In this paper, we introduce branching processes in a L\'evy random
environment. In order to define this class of processes, we study a particular
class of non-negative stochastic differential equations driven by Brownian
motions and Poisson random measures which are mutually independent. The
existence and uniqueness of strong solutions are established under some general
conditions that allows us to consider the case when the strong solution
explodes at a finite time. We use the latter result to construct continuous
state branching processes with immigration and competition in a L\'evy random
environment as a strong solution of a stochastic differential equation. We also
study some properties of such processes that extends recent results obtained by
Bansaye et al. in (Electron. J. Probab. 18, no. 106, 1-31, (2013)), Palau and
Pardo in (arXiv:1506.09197 (2015)) and Evans et al. in (J. Math. Biol., 71,
325-359, (2015)).
