Kramers-Kronig (K-K) relations have constituted one of the principal tools in the optical spectroscopy for the assessment of the optical properties of media from measured spectra. The underlying principle for the existence of the K-K relations is causality. Thanks to the K-K relations we have achieved a better understanding of both macroscopic and microscopic properties of media.Recently, various kinds of modified K-K relations have been presented in the literature. Such relations have been applied, e.g. to the nonlinear optical properties of polymers. A typical advantage of these generalized K-K relations is that the measured data do not need to be manipulated as in the case of the traditional K-K relations. Hence, the accuracy of the inverted data on linear or nonlinear optical properties of media becomes higher.A novel way to utilize generalized K-K relations is related to the measurement and correction of terahertz spectra in the time-domain reflection spectroscopy. Terahertz spectroscopy is nowadays one of the most rapidly developing fields in modern physics with applications being related to, e.g. security at the airports or inspection of pharmaceutical tablets. While recording THz spectra it is also possible to perform a chemical mapping of species. Therefore, correctness of the spectrum is of crucial importance for the identification of different species. This is possible by the generalized K-K relations. In this review paper we consider advances of K-K relations both in nonlinear optical and THz spectroscopy.
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