We consider a system of particles undergoing the branching and annihilating reactions A-->(m+1)A and A+A-->Ø, with m even. The particles move via long-range Lévy flights, where the probability of moving a distance r decays as r(-d-sigma). We analyze this system of branching and annihilating Lévy flights using field theoretic renormalization group techniques close to the upper critical dimension d(c)=sigma with sigma<2. These results are then compared with Monte Carlo simulations in d=1. For sigma close to unity in d=1, the critical point for the transition from an absorbing to an active phase occurs at zero branching. However, for sigma bigger than about 3/2 in d=1, the critical branching rate moves away from zero with increasing sigma, and the transition lies in a different universality class, inaccessible to controlled perturbative expansions. We measure the exponents in both universality classes and examine their behavior as a function of sigma.