In applications to finance, insurance, physics and many other fields, statisticians are often faced with high quality datasets that exhibit deviations from the "normal behavior", caused by the extremes in the sample. As a consequence in recent years a great deal of research has been done in heavy-tailed modelling. Although much of the existing literature focuses on the discrete-time case, the continuous-time heavy-tailed modelling is a very natural technique in many applications and therefore more attention should be paid to the continuous-time case. This is the motivation for the research in this dissertation. We will be focusing mainly on extending the Hill estimator (Hill (1975)) to estimating the tail index of continuous-time stationary stochastic processes. Since one can sample basically as many observations as possible from the continuous-time process, there is a temptation on the practitioner's part to use as large a sample as possible when applying the Hill estimator. We will show that this will lead in many instances to asymptotically inconsistent estimators.