This paper examines persistence, structural breaks and non-linearities in the case of five European stock market indices, namely the FTSE100 (UK), DAX30 (Germany), CAC40 (France), IBEX35 (Spain) and FTSE MIB40 (Italy), using fractional integration methods. The empirical results provide no evidence of non-linearities in either prices or returns; the former are found to exhibit unit roots and the latter to be I(0) in most cases. Further, between 2 and 4 structural breaks are found for each of the return series, and mean reversion in some subsamples. (C) 2020 The Author(s). Published by Elsevier Inc. on behalf of Board of Trustees of the University of Illinois.