Many observed time series processes appear “spiky” due to the occasional appearance of observations particularly large in absolute value. Non-aussian ?-stable distributions, which have regularly varying tail probabilities are often used to model these series. Processes exhibiting non-Gaussian stable behavior have appeared, for example, in economics and finance, signal processing and teletraffic engineering. Parameter estimation for causal, heavy-tailed AR processes has already been considered in the literature (Davis and Resnick (1986), "least squares estimators (LS)"; Davis, Knight and Liu (1992) and Davis (1996), "least absolute deviations (LAD)" and other "M-estimators"; Mikosch, Gadrich, Klüppelberg and Adler (1995), "Whittle estimators" Ling (2005), "weighted least absolute deviations estimators (WLAD)". Theory of convergence has not yet been developed for the distribution of AR param eter estimators when the process is noncausal and heavy-tailed. In this thesis we focus on maximum likelihood (ML) estimation for the parameters of both causal and noncausal autoregressive (AR) time series processes with non-Gaussian stable noise, based on articles by Andrews, Calder and Davis (2009) and Breidt and Davis (1991). Causality (all roots of the AR polynomial are outside the unit circle in the complex plane) is a common assumption in the time series literature, since causal and noncausal models are indistinguishable in the case of Gaussian noise. However, noncausal AR models are identifiable in the case of non-Gaussian noise, and these models are frequently used in deconvolution problems and have also appeared for modeling stock market trading volume data. We, therefore, consider parameter estimation for both causal and noncausal AR models. We assume the parameters of the AR model equation and the parameters of the stable noise distribution are unknown, and we maximize the likelihood function with respect to all parameters. In this thesis, we introduce a nondegenerate limiting distribution for maximum likelihood estimators of the parameters of the autoregressive model equation and the parameters of the stable noise distribution. The estimators for the autoregressive parameters are -consistent and converge in distribution to the maximizer of a random function. The form of this limiting distribution is intractable, but the shape of the distribution for these estimators can be examined using the bootstrap procedure. The bootstrap is asymptotically valid provided the bootstrap sample size ?? with ?0 as n??. The estimators for the parameters of the stable noise distribution have the traditional rate of convergence, asymptotically independent of the AR estimators and have a multivariate normal limiting distribution. The behavior of the estimators for finite samples is studied via simulation by analyzing three autoregressive models with stable noise. These three models are purely noncausal AR(1), noncausal AR(2) and causal AR(1) models.