Long-range dependence (LRD) and non-Gaussianity are ubiquitous in many natural systems such as ecosystems, biological systems and climate. However, it is not always appreciated that the two phenomena may occur together in natural systems and that self-similarity in a system can be a superposition of both phenomena. These features, which are common in complex systems, impact the attribution of trends and the occurrence and clustering of extremes. The risk assessment of systems with these properties will lead to different outcomes (e.g. return periods) than the more common assumption of independence of extremes. Two paradigmatic models are discussed that can simultaneously account for LRD and non-Gaussianity: autoregressive fractional integrated moving average (ARFIMA) and linear fractional stable motion (LFSM). Statistical properties of estimators for LRD and self-similarity are critically assessed. It is found that the most popular estimators can be biased in the presence of important features of many natural systems like trends and multiplicative noise. Also the LRD and non-Gaussianity of two typical natural time series are discussed.