  We introduce the dynamical sine-Gordon equation in two space dimensions with
parameter $\beta$, which is the natural dynamic associated to the usual quantum
sine-Gordon model. It is shown that when $\beta^2 \in (0,\frac{16\pi}{3})$ the
Wick renormalised equation is well-posed. In the regime $\beta^2 \in (0,4\pi)$,
the Da Prato-Debussche method applies, while for $\beta^2 \in
[4\pi,\frac{16\pi}{3})$, the solution theory is provided via the theory of
regularity structures (Hairer 2013). We also show that this model arises
naturally from a class of $2+1$-dimensional equilibrium interface fluctuation
models with periodic nonlinearities.
  The main mathematical difficulty arises in the construction of the model for
the associated regularity structure where the role of the noise is played by a
non-Gaussian random distribution similar to the complex multiplicative Gaussian
chaos recently analysed by Lacoin, Rhodes and Vargas (2013).
