## 1 The Role of Extremes in Society and Scope of the Paper

*Statistics of univariate extremes* (SUE) has been successfully used in the most diverse fields, such as finance, insurance and risk theory, where the *value at risk* at any level *p* (the size of the loss that occurred with small probability *p*) and the adjustment coefficient (a rudimentary measure of risk in a collective of insurance risks) are important parameters of extreme or even rare events. Also, in fields like biology and environment, the Weibull tail coefficient, the regular variation coefficient of the inverse failure rate function, probabilities of exceedance of high levels and endpoints of underlying models (lifetime of human beings or ultimate records in the field of athletics) are relevant extreme events' parameters or functionals. Statistical problems in all these areas have direct ethical, social, economic and environmental impact, and this is one of the reasons that statistics of extremes, in general, and SUE, in particular, have faced a huge development in the last decades. Indeed, rare events can have catastrophic consequences for human activities, through their impact on the natural and constructed environments. The recent development of a sophisticated methodology for the estimation and prediction of functionals of rare events has contributed to saving endangered natural resources and to modelling climate, earthquakes and other environmental phenomena, like precipitation, temperature and floods, situations where we have to deal with large risks or with very low probabilities of overpassing (underpassing) a high (low) level. From a theoretical point of view, the key results obtained by *Fisher & Tippett* 1928 on the possible limiting laws of the sample maxima of a random sample (*X*_{1},…,*X*_{n}) of *independent and identically distributed* (IID) *random variables* (RVs), formalised by *Gnedenko* 1943 and used by *Gumbel* 1958 for applications of *extreme value (EV) theory* (EVT) in engineering subjects, are some of the key tools that led to the way statistical EVT has been exploding in the last decades. The statistical applications of EVT gave emphasis to the relaxation of the independence condition, to the consideration of multivariate and spatial frameworks and to an increasing use of regular variation and point process approaches. These topics are well documented in books by *David* 1970, *Galambos* 1978, *Leadbetter et al.*, 1983, *Resnick* 1987, *Arnold et al.*, 1992, *Falk et al.*, 1994, *Embrechts et al.*, 1997, *Reiss & Thomas* 1997, *Coles* 2001, *David & Nagaraja* 2003, *Beirlant et al.*, 2004, *Castillo et al.*, 2005, *de Haan & Ferreira* 2006, *Resnick* 2007, *Markovich* 2007 and their subsequent editions. For an overview of most of the topics in this field, see the recent volumes of *Extremes* 11:1 (2008) and *Revstat* 10:1 (2012). Among review papers in the area of SUE, we mention Gomes *et al.*, 2008a, 2007b, *Neves & Fraga Alves* 2008, *Hüsler & Peng* 2008, *Beirlant et al.*, 2012 and *Scarrot & McDonald* 2012.

In Section 2 of this review paper, we provide some details related to the non-degenerate limiting behaviour of the sequence of maximum values, other top order statistics (OSs) and excesses over high thresholds. Section 3 is dedicated to the most common parametric models in SUE and to a discussion of the more recent semi-parametric frameworks. Estimation procedures are discussed in Sections 4, 5 and 6. In Section 7, we briefly discuss SUE for censored data. Section 8 is dedicated to a brief reference to testing issues either under parametric or semi-parametric frameworks. In Section 9, we make a brief reference to the estimation of the extremal index. Finally, in Section 10, we mention a few other relevant topics in the area and a few important open problems that we think useful to be dealt with in the nearby future.