Is there "a rate of convergence" for the central limit theorem?

Question

Let $X_i$ be i.i.d. with zero mean and unit variance. Then according to the central limit theorem $$\frac{1}{\sqrt{n}} \sum \limits_{i=1}^n X_i \to_D \mathcal{N}(0,1),$$ as $n \to \infty$, where $\to_D$ denotes convergence in distribution. Can you say anything about the finite sample properties of the left hand side? Ideally, I would like to say in precise way that the distribution of the LHS is "almost" the RHS.