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  1. Multi-phase filamentary structures around brightest cluster galaxies (BCG) are likely a key step of AGN-feedback. We observed molecular gas in three cool cluster cores, namely Centaurus, Abell S1101, and RXJ1539.5, and gathered ALMA (Atacama Large Millimeter/submillimeter Array) and MUSE (Multi Unit Spectroscopic Explorer) data for 12 other clusters. Those observations show clumpy, massive, and long (3−25 kpc) molecular filaments, preferentially located around the radio bubbles inflated by the AGN. Two objects show nuclear molecular disks. The optical nebula is certainly tracing the warm envelopes of cold molecular filaments. Surprisingly, the radial profile of the H α /CO flux ratio is roughly constant for most of the objects, suggesting that (i) between 1.2 and 6 times more cold gas could be present and (ii) local processes must be responsible for the excitation. Projected velocities are between 100 and 400 km s −1 , with disturbed kinematics and sometimes coherent gradients. This is likely due to the mixing in projection of several thin (and as yet) unresolved filaments. The velocity fields may be stirred by turbulence induced by bubbles, jets, or merger-induced sloshing. Velocity and dispersions are low, below the escape velocity. Cold clouds should eventually fall back and fuel the AGN.more »We compare the radial extent of the filaments, r fil , with the region where the X-ray gas can become thermally unstable. The filaments are always inside the low-entropy and short-cooling-time region, where t cool / t ff  <  20 (9 of 13 sources). The range of t cool / t ff of 8−23 at r fil , is likely due to (i) a more complex gravitational potential affecting the free-fall time t ff (sloshing, mergers, etc.) and (ii) the presence of inhomogeneities or uplifted gas in the ICM, affecting the cooling time t cool . For some of the sources, r fil lies where the ratio of the cooling time to the eddy-turnover time, t cool / t eddy , is approximately unity.« less