In this paper, we propose a Double Thompson Sampling (D-TS) algorithm for dueling bandit problems. As its name suggests, D-TS selects both the first and the second candidates according to Thompson Sampling. Specifically, D-TS maintains a posterior distribution for the preference matrix, and chooses the pair of arms for comparison according to two sets of samples independently drawn from the posterior distribution. This simple algorithm applies to general Copeland dueling bandits, including Condorcet dueling bandits as its special case. For general Copeland dueling bandits, we show that D-TS achieves $O(K^2 \log T)$ regret. Moreover, using a back substitution argument, we refine the regret to $O(K \log T + K^2 \log \log T)$ in Condorcet dueling bandits and many practical Copeland dueling bandits. In addition, we propose an enhancement of D-TS, referred to as D-TS+, that reduces the regret by carefully breaking ties. Experiments based on both synthetic and real-world data demonstrate that D-TS and D-TS$^+$ significantly improve the overall performance, in terms of regret and robustness.