Modelling heavy tails and double long memory in stock returns is very important for financial asset pricing, asset allocation and risk management. In this paper, we demonstrate that an a- stable distribution is better fitted to the North African stock return data in TUNINDEX (Tunisia), MASI (Morocco) and EGX30 (Egypt) than the normal distribution. The empirical results show that the asymmetric leptokurtic features presented in these markets can be captured by an a-stable distribution. Moreover, estimation of the tail index allows us to determine the long-memory behaviour of stock returns. Additionally, this study examines the long-memory property in mean returns and volatility of these markets. The results indicate that long-memory dynamics in the returns and volatility might be modelled by the joint ARFIMA–FIGARCH model. The results of the joint ARFIMA–FIGARCH model show strong evidence of long memory in both returns and volatility. The long memory in returns implies that stock prices follow a predictable behaviour, which is inconsistent with the efficient market hypothesis. The evidence of long memory in volatility, however, shows that uncertainty or risk is an important determinant of the behaviour of daily stock data in North African stock markets. The implication of the present work is that the assumption of non-normality provides better specifications regarding the long-memory property.