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This paper introduces an extension of the linear least-squares (orLomb-Scargle) periodogram for the case when the model of the signal to bedetected is non-sinusoidal and depends on unknown parameters in a non-linearmanner. The attention is paid to the problem of estimating the statisticalsignificance of candidate periodicities found using such non-linearperiodograms. This problem is related to the task of quantifying thedistributions of maximum values of these periodograms. Based on recent resultsin the mathematical theory of extreme values of random field (the generalizedRice method), we give a general approach to find handy analytic approximationfor these distributions. This approximation has the general form $e^{-z}P(\sqrt z)$, where $P$ is an algebraic polynomial and $z$ being the periodogrammaximum.  The general tools developed in this paper can be used in a wide variety ofastronomical applications, for instance in the studies of variable stars andextrasolar planets. For this goal, we develop and consider in details theso-called von Mises periodogram: a specialized non-linear periodogram where thesignal is modelled by the von Mises periodic function $\exp(\nu \cos \omegat)$. This simple function with an additional non-linear parameter $\nu$ canmodel lightcurves of many astronomical objects that show periodic photometricvariability of different nature. We prove that our approach can be perfectlyapplied to this non-linear periodogram.  We provide a package of auxiliary C++ programs, attached as the online-onlymaterial. They should faciliate the use of the von Mises periodogram inpractice.

Detecting non-sinusoidal periodicities in observational data: the von Mises periodogram for variable stars and exoplanetary transits