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In a range of machine learning applications, from medical diagnostics to causality, distributions appear as inputs, and these distributions might not even be directly accessible, but only through samples thereof. Kernel-based methods are well-suited for this problem setting by first embedding the distributions into a Hilbert space and then using a standard kernel on this Hilbert space. This strategy has received particular attention for the case of regression with distributions as inputs, called distributional regression, and by now efficient algorithms as well as substantial theory are available. However, for other variants of learning with distributional inputs there has been much less work, most notably for the case of distributional classification. Motivated by this latter fact, we present recent work on advancing statistical learning theory for kernel methods with distributional inputs, covering important settings like binary classification with support vector machines.