Two families of dependence measures between random variables are introduced. They are based on the Rényi divergence of order $α$ and the relative $α$-entropy, respectively, and both dependence measures reduce to Shannon's mutual information when their order $α$ is one. The first measure shares many properties with the mutual information, including the data-processing inequality, and can be related to the optimal error exponents in composite hypothesis testing. The second measure does not satisfy the data-processing inequality, but appears naturally in the context of distributed task encoding.