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This study sought to model the stock market return volatility at the Nairobi Securities Exchange (NSE) in the presence of structural breaks. Using daily NSE 20 share index for the period 04/01/2010  to  29/12/2017,  the market return volatility was modeled using different GARCH type models and taking into account four endogenously identified structural breaks. The market exhibited a non-normal distribution that was leptokurtic and negatively skewed and also showed evidence for ARCH effects, volatility clustering, and volatility persistence. We found that by considering structural breaks, volatility persistence was reduced, while leverage effects were found to lead to explosive volatility. In addition, investors were not rewarded for taking up additional risk since the risk premium was insignificant for the full period. However, during explosive volatility, investors were rewarded for taking up more risk. Moreover, we found that risk premium, leverage effects, and volatility persistence were significantly correlated. The GARCH (1,1) and TGARCH(1,1) models were found to be the best fit models to test for symmetric and asymmetric effects respectively. While the GARCH models were able to provide evidence for the stylized facts in the NSE, we conclude that the presence or absence of these features is period specific. This especially relates to volatility persistence, leverage effects, and risk premium effects. Caution should, therefore, be taken in using a specific GARCH model to forecast market return volatility in Kenya. It is thus imperative to pretest the data before any return volatility forecasting is done.