The Baker-Campbell-Hausdorff (BCH) series is a very prominent subject in Lie theory. It is useful for fundamental research as well as for applications, e.g., for the numerical treatment of differential equations on manifolds. Consider two noncommutative variables \(X,Y\). Then the BCH series is the formal power series of \(\log(e^Xe^Y)\). Considering a finite-dimensional matrix Lie algebra, the question is, under which conditions the BCH series converges. It is well-known that, in general, the BCH series does not converge globally, while it converges for elements \(X,Y\) sufficiently close to \(1\). The authors collect various estimates appearing in the literature for the convergence domain and improve all of them by a new one, which enlarges the already known estimates significantly. Moreover, they point out that if the commutator of \(X\) and \(Y\) is sufficiently small, \(X+Y\) is in some sense a good estimate for \(\log(e^Xe^Y)\) even if the BCH series does not converge. The paper is completed by considerations on an efficient way to compute the BCH series approximately.