This paper is concerned with the stability of analytical and numerical solutions for nonlinear stochastic delay differential equations (SDDEs) with jumps. A sufficient condition for mean-square exponential stability of the exact solution is derived. Then, mean-square stability of the numerical solution is investigated. It is shown that the compensated stochastic θ methods inherit stability property of the exact solution. More precisely, the methods are mean-square stable for any stepsize $\Delta t=\tau /m$ when $1/2\le \theta \le 1$, and they are exponentially mean-square stable if the stepsize $\Delta t\in (0,\Delta t_0)$ when . Finally, some numerical experiments are given to illustrate the theoretical results.