Summary: For fairly broad classes of topological vector spaces \(X\) and \(Y\), there are given complete characterizations of those maps \(f\:X\to Y\) for which the induced transformation of series \(\sum _n x_n\rightsquigarrow \sum _n f(x_n)\) preserves properties such as convergence, boundedness, absolute convergence, and unconditional convergence. For example, the following extension of a result of R. Rado for the case of normed spaces is shown: If \(X\) is metrizable, and \(Y\) is sequentially complete and contains no isomorphic copy of the space \(\omega \) of all scalar sequences, then \(f\) preserves convergence of series if and only if \(f\) is additive and continuous in a neighborhood of zero.