In recent years there has been an increasing interest in learning Bayesian networks from data. One of the most effective methods for learning such networks is based on the minimum description length (MDL) principle. Previous work has shown that this learning procedure is asymptotically successful: with probability one, it will converge to the target distribution, given a sufficient number of samples. However, the rate of this convergence has been hitherto unknown. In this work we examine the sample complexity of MDL based learning procedures for Bayesian networks. We show that the number of samples needed to learn an $\epsilon$-close approximation (in terms of entropy distance) with confidence $\delta$ is $O\left((\frac{1}{\epsilon})^{\frac{4}{3}}\log\frac{1}{\epsilon} \log\frac{1}{\delta}\log\log\frac{1}{\delta}\right)$. This means that the sample complexity is a low-order polynomial in the error threshold and sub-linear in the confidence bound. We also discuss how the constants in this term depend on the complexity of the target distribution. Finally, we address questions of asymptotic minimality and propose a method for using the sample complexity results to speed up the learning process.