We study the complexity of the satisfiability problem for a class of logical formulae called iterated propositional schemata, modelling infinite sequences of structurally similar propositional formulae. We prove that the problem is EXPSPACE-complete in general and PSPACE-complete if the numbers occurring in the formula are polynomially bounded by the size of the schema. We then consider more restricted classes: we prove that the problem is still PSPACE-complete for the Horn class, but only NP-complete for the Krom class (sets of clauses of length 2). Finally, we devise a simple criterion ensuring that the satisfiability problem is in P.