Since 2014 the layered semiconductor SnSe in the high-temperature Cmcm phase is known to be the most efficient thermoelectric material. Making use of first-principles calculations we show that its vibrational and thermal transport properties are determined by huge non-perturbative anharmonic effects. We show that the transition from the Cmcm phase to the low-symmetry P nma is a second-order phase transition driven by the collapse of a zone border phonon, whose frequency vanishes at the transition temperature. Our calculations show that the spectral function of the in-plane vibrational modes are strongly anomalous with shoulders and double-peak structures. We calculate the lattice thermal conductivity obtaining good agreement with experiments only when non-perturbative anharmonic scattering is included. Our results suggest that the good thermoelectric efficiency of SnSe is strongly affected by the non-perturbative anharmonicity. We expect similar effects to occur in other thermoelectric materials.Thermoelectric materials can convert waste heat into useful electricity 1,2 . The thermoelectric efficiency of a material is measured by the dimensionless figure of merit ZT = S 2 σT /κ, where S is the Seebeck coefficient, σ the electrical conductivity, T the temperature, and κ = κ e + κ l the thermal conductivity, constituted by electronic κ e and lattice κ l contributions. The thermoelectric efficiency can be thus enhanced by decreasing the thermal conductivity while keeping a high power factor S 2 σ. Materials have been doped 3-5 or nanostructured 6,7 in order to get a high power factor combined with a low thermal conductivity, yielding, i.e., ZT 2.2 in PbTe 8 . In the proximity to a phase transition ZT may also soar, as in the case of Cu 2 Se 9 . Recently, however, Zhao et al. reported for SnSe 10 the highest thermoelectric figure of merit ever reached in a material without doping, material treatment or without being in the proximity to a phase transition: ZT 2.6 above 800 K.SnSe is a narrow gap semiconductor that crystallizes at room temperature in an orthorhombic P nma phase. At T 800 K 10-13 it transforms into a more symmetric base-centered orthorhombic Cmcm structure (see Fig. 1). The order of the transition is not clear: some works 10-12 claim it is a continuous second-order transition and others it has a first-order character 13 . A recent work 14 argues the transition occurs in two steps, where increasing temperature induces first a change in the lat-tice parameters that induces after a lattice instability. There is no inelastic scattering experiment so far for the high-temperature phase, which should show a prominent phonon collapse at the transition temperature if it belonged to the displacive second-order type 15-17 .The most interesting thermoelectric properties appear in the high-temperature Cmcm phase, where the reduction of the electronic band gap increases the number of carriers providing a higher power factor, while the thermal conductivity remains very low 10 . The value of the intrinsic κ l of SnSe r...