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Extreme weather and climate events, although rare at any particular location, can lead to different amount of loss to exposed human and natural systems, even to disasters. In this paper, the most authoritative definitions of "extreme events" given by the World Meteorological Organization and Intergovernmental Panel on Climate Change have been considered, as well as the underlying basic concepts, i.e., selected intensity levels, selected percentiles, multiples of the standard deviation, return period, distribution tails, imprint left, cause-effect relationships, natural disasters. Definitions and criteria have been tested with real world case studies using long instrumental records (300 years of daily temperature and 200 years of daily precipitation in Bologna, Italy) and proxy series (1000 years of Venice lagoon frozen over and 300 years of Po River outflow). The analysis reveals that each definition leads to particular consequences, e.g., in the peak over threshold theory, if the threshold is expressed in absolute terms, the number of extreme events may change with the climate period as tested with the Venice case study; as opposed, the relative definition based on percentiles or standard deviation will keep unchanged this frequency. Again, considering extreme events those external to the 10th or 90th percentile of the distribution may lead to a return period too short, e.g., 10 days for daily records, while a 10 years return period would require 1st or 99th percentiles, as tested with the daily temperature and precipitation in Bologna. In addition, the distribution of a series may substantially change shape passing from daily to monthly and yearly averages as tested with the series taken as examples. The specific case of proxies is also considered analysing their uncertainties and categorization. The lagoon frozen over as a consequence of exceptionally severe winters constitutes an example of extremes based on an absolute threshold, as well as cause-effect relationship, and the return period was highly affected by the change of climate periods. The example of the overflow of the Po River suggests that the occurrence of extremes, and their intensity, may be altered by other factors that concur to the final result. [ABSTRACT FROM AUTHOR]