It is well known to applied statisticians and scientists that the assumption of independence is often not valid for real data. In particular, even when all precautions are taken to prevent dependence, slowly decaying serial correlations frequently occur. If not taken into account, they can have disastrous effects on statistical inference. This phenomenon has been observed empirically by many prominent scientists long before suitable mathematical models were known. Apart from some scattered early references, mathematical models with long-range dependence were first introduced to statistics by Mandelbrot and his co-workers (Mandelbrot and Wallis, 1968, 1969; Mandelbrot and van Ness, 1968). Since then, long-range dependence in statistics has gained increasing attention. Parsimonious models with long memory are stationary increments of self-similar processes with self-similarity parameter $H \in (1/2,1)$, fractional ARIMA processes and other stationary stochastic processes with non-summable correlations. In the last decade, many results on statistical inference for such processes have been established. In the present paper, a review of these results is given.