This thesis contains five essays on fractional cointegration, seasonal fractional cointegration and seasonal long memory. After an introduction in the first Chapter, Chapter 2 reviews competing tests for fractional cointegration, since no standard approach emerged so far. It provides a synthesis of the literature and a detailed comparative Monte Carlo study to guide empirical researchers in their choice of appropriate methodologies. In Chapter 3, the previously reviewed methods are applied in the context of the European government bond market analyzing the degree of market integration.Chapter 4 deals with possible breaks in the persistence structure of a fractional cointegrating relationship. It introduces test procedures for no fractional cointegration that are robust for such a break. The following Chapters 5 and 6 consider another phenomenon in time series, namely seasonality, in particular seasonal long memory. Chapter 5 examines multivariate seasonal data and the concomitant possibility of seasonal fractional cointegration. Chapter 6 takes a different perspective and deals with univariate seasonal time series and proposes a test for seasonal long memory with a known frequency is proposed.