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Classification results for easy quantum groups

We study the orthogonal quantum groups satisfying the ``easiness'' assumption axiomatized in our previous paper, with the construction of some new examples, and with some partial classification results. The conjectural conclusion is that the easy quantum groups consist of the previously known 14 examples, plus of an hypothetical multi-parameter ``hyperoctahedral series'', related to the complex reflection groups $H_n^s=\mathbb Z_s\wr S_n$. We discuss as well the general structure, and the computation of asymptotic laws of characters, for the new quantum groups that we construct.