We study the stochastic heat equation on the real line

$$\begin{equation*}\frac{\partial u}{\partial t} = \frac12 \frac{\partial^2 u}{\partial x^2} + \sigma(u) \dot{W}\end{equation*}$$

(where $\dot{W}$ is a space time white noise and $\sigma$ is differentiable with a bounded derivative). The main result of this talk is that: the spatial integral $\int_{-R}^R u(t,x) dx$ converges to a Gaussian distribution as $R \to \infty$, after renormalization. It is proved using Stein's method and Malliavin calculus, which will be introduced in the talk. This result is based on a joint work with Nualart and Viitasaari.