This article reports the absolute and convective instability of an annular viscous liquid jet with its inner and outer surfaces exposed to in viscid gas streams of unequal velocities. The effects of geometric and various flow parameters are investigated. It is found tbatboth absolute and connective instability exist for pares-sinuous and -varicose modes under certain flow conditions. For pares-sinuous mode, the annular liquid jet, with an inner gas moving at relatively small velocities, can have either convective or absolute instability, depending on specific flow conditions. However, the jet is of only absolute instability if the innergas is stationary or moves at sufficiently large velocities. Pares-sinuous unstable waves outgrow para-varicose waves and hence dominate the jet instability process, according to both absolute and convective instability analyses. The liquid viscosity has a stabilizing effect on the jet instability, but the gas inertial force shows a fairly complex influence on the absolute instability. The convective growth rates for various inner gas velocities indicate that not only the velocity difference between, but also the absolute velocity of the liquid and gas, determine the jet breakup process.