In recent years, there is growing evidence that long-term correlations play an important role in climate, physiology, and computer science as well as in financial markets; the examples range from river floods, temperatures, and wind fields to market volatilities, heart-beat intervals and internet traffic. Here, we review several methods to detect long-term correlations, also in the presence of external trends, and give examples for long-term correlations in climate. Finally, we show how external trends can be detected in data with long-term memory, and apply the results to estimate the anthropogeneous part of the recent warming of the athmosphere and the oceans.