This thesis examines issues of seasonality in ARMA processes, comparing and contrasting Box-Jenkins seasonality with Gegenbauer seasonality, and the SGAR or seasonal GAR model. Novel contributions of this thesis include new estimation methods for Gegenbauer seasonality - a log-Whittle estimator and a Bayesian MCMC estimator using a Whittle likelihood, and also the Seasonal GAR model which is compared to the Box-Jenkins and the Gegenbauer seasonality models. Partial consistency and distributional attributes of the parameters of a Gegenbauer process when estimated by a log-Whittle estimator are presented along with distributional results for the log-periodogram of a Gegenbauer process. Consistency and distributional results are also presented for the novel seasonal GAR (SGAR) model. Monte Carlo simulations are evaluated for all the methods listed above to illustrate which circumstances might favour one or the other methods.