Let u be the solution to the one-dimensional stochastic heat equation driven by a space-time white noise with constant initial condition. The purpose of this talk is to present a recent result on the uniform convergence of the density of the normalized spatial averages of the solution u on an interval [−R,R], as R tends to infinity, to the density of the standard normal distribution, assuming some non-degeneracy and regularity conditions on the diffusion coefficient. These results are based on the combination of Stein's method for normal approximations and Malliavin calculus techniques. This is a joint work with David Nualart.