Rademacher (Mathematische Annalen87 (1922) 112–138), Steinhaus (Mathematische Zeitschrift31 (1930) 408–416) and Paley and Zygmund (Mathematical Proceedings of the Cambridge Philosophical Society26 (1930) 337–257, Mathematical Proceedings of the Cambridge Philosophical Society26 (1930) 458–474, Mathematical Proceedings of the Cambridge Philosophical Society28 (1932) 190–205) initiated the extensive study of random series. Using the theory of algorithmic randomness, which is a mix of computability theory and probability theory, we investigate the effective content of some classical theorems. We discuss how this is related to an old question of Kahane and Bollobás. We also discuss how considerations of such algorithmic questions about random series seem to lead to new notions of algorithmic randomness.