  We use first principles calculations to study structural, vibrational and
superconducting properties of H$_2$S at pressures $P\ge 200$ GPa. The inclusion
of zero point energy leads to two different possible dissociations of H$_2$S,
namely 3H$_2$S $\to$ 2H$_3$S + S and 5H$_2$S $\to$ 3H$_3$S + HS$_2$, where both
H$_3$S and HS$_2$ are metallic. For H$_3$S, we perform non-perturbative
calculations of anharmonic effects within the self-consistent harmonic
approximation and show that the harmonic approximation strongly overestimates
the electron-phonon interaction ($\lambda\approx 2.64$ at 200 GPa) and T$_c$.
Anharmonicity hardens HS bond-stretching modes and softens H--S bond-bending
modes. As a result, the electron-phonon coupling is suppressed by $30\%$
($\lambda\approx 1.84$ at 200 GPa). Moreover, while at the harmonic level T$_c$
decreases with increasing pressure, the inclusion of anharmonicity leads to a
T$_c$ that is almost independent of pressure. High pressure hydrogen sulfide is
a strongly anharmonic superconductor.
