This thesis examines some empirical applications of long memory processes with specific interest in the future of crude oil prices market in Oman. This is done in three parts. The first part is the conditional mean of long memory in the crude oil Market using monthly data. The empirical results provide strong support for long memory in the crude oil market of Oman in line with previous studies. The second part is the conditional volatility modelling of different convenience volatility models for modelling the volatility of crude oil market in Oman. The implied convenience yield for crude oil market is found to be driven by the component standard GARCH model. The third part is the conditional distribution of the errors. The result provided strong evidence that assuming a heavier tailed error distribution than the normal distribution and modelling the conditional mean using long memory models improves the fit. The focus of this thesis is in the comparison of the volatility forecasting performance of five commonly used forecasting models, namely: the standard-GARCH - ARFIMA model, the integrated-GARCH - ARFIMA model, the gjr-GARCH - ARFIMA model, the exponential-GARCH - ARFIMA model and the component S-GARCH - ARFIMA model, with seven different conditional distributions. These are the generalized error distribution, the normal distribution, the skew-student distribution, the Johnson's reparametrized SU distribution, the generalized hyperbolic distribution, the normal inverse Gaussian distribution and the GH skew-student distribution. The dataset used in this project is monthly data for Oman crude oil prices (by dollar per barrel) from November 2002 until January 2017. The main objective of this project is to compare the volatility models in terms of the in-sample and out-of-sample fit. Moreover a main conclusion is that yes, the more complex models do provide a better in sample fit than the more parsimonious models. However in terms of the out-of-sample forecasting performance the result was conclusive using mean square error as a forecast evaluation criteria. The best model found to be ARFIMA (2, 0.143302,2)-csGARCH(1,1) with skewed student distribution conditional distribution model and it is recommended to be used for one step ahead forecast. An important finding is using Brent oil prices as an external regressor. This improves our model in terms of in sample forecast. However, it is not necessarily that the model with the best in-sample fit produces the best out-of-sample forecast.