We design new differentially private algorithms for the problems of adversarial bandits and bandits with expert advice. For adversarial bandits, we give a simple and efficient conversion of any non-private bandit algorithm to a private bandit algorithm. Instantiating our conversion with existing non-private bandit algorithms gives a regret upper bound of $O\left(\frac{\sqrt{KT}}{\sqrtε}\right)$, improving upon the existing upper bound $O\left(\frac{\sqrt{KT \log(KT)}}ε\right)$ for all $ε\leq 1$. In particular, our algorithms allow for sublinear expected regret even when $ε\leq \frac{1}{\sqrt{T}}$, establishing the first known separation between central and local differential privacy for this problem. For bandits with expert advice, we give the first differentially private algorithms, with expected regret $O\left(\frac{\sqrt{NT}}{\sqrtε}\right), O\left(\frac{\sqrt{KT\log(N)}\log(KT)}ε\right)$, and $\tilde{O}\left(\frac{N^{1/6}K^{1/2}T^{2/3}\log(NT)}{ε^{1/3}} + \frac{N^{1/2}\log(NT)}ε\right)$, where $K$ and $N$ are the number of actions and experts respectively. These rates allow us to get sublinear regret for different combinations of small and large $K, N$ and $ε.$