The development of free probability theory has drawn much inspiration from its deep and far reaching analogy with classical probability theory. The same holds for its operator-valued extension, where the fundamental notion of free independence is generalized to free independence with amalgamation as a kind of conditional version of the former. Its development naturally led to operator-valued free analogues of key and fundamental limiting theorems such as the operator-valued free Central Limit Theorem due to Voiculescu and results about the asymptotic behaviour of distributions of matrices with operator-valued entries. In this talk, we show Berry-Esseen bounds for such limit theorems. The estimates are on the level of operator-valued Cauchy transforms and the L{\'e}vy distance. We address also the multivariate setting for which we consider linear matrix pencils and noncommutative polynomials as test functions. The estimates are in terms of operator-valued moments and yield the first quantitative bounds on the L{\'e}vy distance for the operator-valued free CLT. This also yields quantitative estimates on joint noncommutative distributions of operator-valued matrices having a general covariance profile. This is a joint work with Tobias Mai.